academic excellence in physics for lecturers
TRANSCRIPT
Academic Excellence in PHYSICS
for Lecturers
April 2015
© SCERT
April 2015
250 Copies
Chief Advisor
Anita Satia
Director, SCERT
Guidance
Dr. Pratibha Sharma
Joint Director, SCERT
Academic Co-ordinator and Editor
Sapna Yadav
Sr. Lecturer, SCERT
Contributors
Pundrikaksh Kaundinya, Principal, RPVV, Rajniwas Marg
Devendra Kumar, Vice Principal, SBBM SV Shankracharya Marg, Delhi -54
P.N Varshney, Retd. Principal, President, Delhi
R.RangaRajan, PGT Physics, DTEA School
Sapna Yadav, Sr. Lecturer, SCERT
Publication Officer
Ms. Sapna Yadav
Publication Team
Sh. Navin Kumar, Ms. Radha, Sh. Jai Baghwan
Published by : State Council of Educational Research & Training, New Delhi & printed at
Graphic Printers, Karol Bagh, New Delhi
CONTENTS
S.No. Content Page no.
1. Syllabus - PHYSICS (Code No. 042) 1-19
2. Delhi State Science Teachers Forum 20-21
3. CBSE Board Paper 2014-15 with Solution 27-45
4. Changes in Physics Syllabus 45
5. Supplementary Material on Additional Sub-Topics included in Class XI and XII Physics Theory Syllabus
46-60
6. Unit 1 : Units, Dimensions and Measurement
61-67
7. Unit 2 : Kinetics 68-88
8. Unit 3 : Newton's Law of Motion 89-103
9. Unit 4 : Work, Energy and Power 104-111
10. Unit 5 : Rotational Motion 112-124
11. Unit 6: Gravitation 125-133
12. Unit 7 : Properties of Matter 134-150
13. Unit 8 : Simple Harmonic Motion 151-158
14. Unit 9 : Wave Motion 159-168
15. Appendix 169-
PHYSICS (Code No. 042)
Senior Secondary stage of school education is a stage of transition from general education to discipline-based
focus on curriculum. The present updated syllabus keeps in view the rigour and depth of disciplinary
approach as well as the comprehension level of learners. Due care has also been taken that the syllabus is
comparable to the international standards. Salient features of the syllabus include:
Emphasis on basic conceptual understanding of the content.
Emphasis on use of SI units, symbols, nomenclature of physical quantities and formulations as per
international standards.
Providing logical sequencing of units of the subject matter and proper placement of concepts with their
linkage for better learning.
Reducing the curriculum load by eliminating overlapping of concepts/content within the discipline and
other disciplines.
Promotion of process-skills, problem-solving abilities and applications of Physics concepts.
Besides, the syllabus also attempts to
strengthen the concepts developed at the secondary stage to provide firm foundation for further
learning in the subject.
expose the learners to different processes used in Physics-related industrial and technological
applications.
develop process-skills and experimental, observational, manipulative, decision making and investigatory
skills in the learners.
promote problem solving abilities and creative thinking in learners.
develop conceptual competence in the learners and make them realize and appreciate the interface of
Physics with other disciplines.
PHYSICS (Code No. 042)
COURSE STRUCTURE
Class XI (Theory) (2015-16)
Time: 3 hrs. Max Marks: 70
No. of Periods Marks
Unit–I Physical World and Measurement
10
23
Chapter–1: Physical World
Chapter–2: Units and Measurements
Unit-II Kinematics
24 Chapter–3: Motion in a Straight Line
Chapter–4: Motion in a Plane
Unit–III Laws of Motion 14
Chapter–5: Laws of Motion
1
Unit–IV Work, Energy and Power 12
17
Chapter–6: Work, Energy and Power
Unit–V Motion of System of Particles and Rigid Body 18
Chapter–7: System of Particles and Rotational Motion
Unit-VI Gravitation 12
Chapter–8: Gravitation
Unit–VII Properties of Bulk Matter
24
20
Chapter–9: Mechanical Properties of Solids
Chapter–10: Mechanical Properties of Fluids
Chapter–11: Thermal Properties of Matter
Unit–VIII Thermodynamics 12
Chapter–12: Thermodynamics
Unit–IX Behaviour of Perfect Gases and Kinetic Theory of
Gases 08
Chapter–13: Kinetic Theory
Unit–X Oscillations and Waves
26 10 Chapter–14: Oscillations
Chapter–15: Waves
Total 160 70
Unit I: Physical World and Measurement 10 Periods
Chapter–1: Physical World
Physics-scope and excitement; nature of physical laws; Physics, technology and society.
Chapter–2: Units and Measurements
Need for measurement: Units of measurement; systems of units; SI units, fundamental and
derived units. Length, mass and time measurements; accuracy and precision of measuring
instruments; errors in measurement; significant figures.
Dimensions of physical quantities, dimensional analysis and its applications.
Unit II: Kinematics 24 Periods
Chapter–3: Motion in a Straight Line
Frame of reference, Motion in a straight line: Position-time graph, speed and velocity.
Elementary concepts of differentiation and integration for describing motion, uniform and non-
uniform motion, average speed and instantaneous velocity, uniformly accelerated motion,
velocity - time and position-time graphs.
Relations for uniformly accelerated motion (graphical treatment).
2
Chapter–4: Motion in a Plane
Scalar and vector quantities; position and displacement vectors, general vectors and their
notations; equality of vectors, multiplication of vectors by a real number; addition and
subtraction of vectors, relative velocity, Unit vector; resolution of a vector in a plane,
rectangular components, Scalar and Vector product of vectors.
Motion in a plane, cases of uniform velocity and uniform acceleration-projectile motion,
uniform circular motion.
Unit III: Laws of Motion 14 Periods
Chapter–5: Laws of Motion
Intuitive concept of force, Inertia, Newton's first law of motion; momentum and Newton's
second law of motion; impulse; Newton's third law of motion.
Law of conservation of linear momentum and its applications.
Equilibrium of concurrent forces, Static and kinetic friction, laws of friction, rolling friction,
lubrication.
Dynamics of uniform circular motion: Centripetal force, examples of circular motion (vehicle on
a level circular road, vehicle on a banked road).
Unit IV: Work, Energy and Power 12 Periods
Chapter–6: Work, Engery and Power
Work done by a constant force and a variable force; kinetic energy, work-energy theorem,
power.
Notion of potential energy, potential energy of a spring, conservative forces: conservation of
mechanical energy (kinetic and potential energies); non-conservative forces: motion in a
vertical circle; elastic and inelastic collisions in one and two dimensions.
Unit V: Motion of System of Particles and Rigid Body 18 Periods
Chapter–7: System of Particles and Rotational Motion
Centre of mass of a two-particle system, momentum conservation and centre of mass motion.
Centre of mass of a rigid body; centre of mass of a uniform rod.
Moment of a force, torque, angular momentum, laws of conservation of angular momentum and
its applications.
Equilibrium of rigid bodies, rigid body rotation and equations of rotational motion, comparison
of linear and rotational motions.
Moment of inertia, radius of gyration, values of moments of inertia for simple geometrical
objects (no derivation). Statement of parallel and perpendicular axes theorems and their
applications.
Unit VI: Gravitation 12 Periods
Chapter–8: Gravitation
Kepler's laws of planetary motion, universal law of gravitation.
Acceleration due to gravity and its variation with altitude and depth.
3
Gravitational potential energy and gravitational potential, escape velocity, orbital velocity of a
satellite, Geo-stationary satellites.
Unit VII: Properties of Bulk Matter 24 Periods
Chapter–9: Mechanical Properties of Solids
Elastic behaviour, Stress-strain relationship, Hooke's law, Young's modulus, bulk modulus, shear
modulus of rigidity, Poisson's ratio; elastic energy.
Chapter–10: Mechanical Properties of Fluids
Pressure due to a fluid column; Pascal's law and its applications (hydraulic lift and hydraulic
brakes), effect of gravity on fluid pressure.
Viscosity, Stokes' law, terminal velocity, streamline and turbulent flow, critical velocity,
Bernoulli's theorem and its applications.
Surface energy and surface tension, angle of contact, excess of pressure across a curved
surface, application of surface tension ideas to drops, bubbles and capillary rise.
Chapter–11: Thermal Properties of Matter
Heat, temperature, thermal expansion; thermal expansion of solids, liquids and gases,
anomalous expansion of water; specific heat capacity; Cp, Cv - calorimetry; change of state -
latent heat capacity.
Heat transfer-conduction, convection and radiation, thermal conductivity, qualitative ideas of
Blackbody radiation, Wein's displacement Law, Stefan's law, Green house effect.
Unit VIII: Thermodynamics 12 Periods
Chapter–12: Thermodynamics
Thermal equilibrium and definition of temperature (zeroth law of thermodynamics), heat, work
and internal energy. First law of thermodynamics, isothermal and adiabatic processes.
Second law of thermodynamics: reversible and irreversible processes, Heat engine and
refrigerator.
Unit IX: Behaviour of Perfect Gases and Kinetic Theory of Gases 08 Periods
Chapter–13: Kinetic Theory
Equation of state of a perfect gas, work done in compressing a gas.
Kinetic theory of gases - assumptions, concept of pressure. Kinetic interpretation of
temperature; rms speed of gas molecules; degrees of freedom, law of equi-partition of energy
(statement only) and application to specific heat capacities of gases; concept of mean free
path, Avogadro's number.
Unit X: Oscillations and Waves 26 Periods
Chapter–14: Oscillations
Periodic motion - time period, frequency, displacement as a function of time, periodic functions.
Simple harmonic motion (S.H.M) and its equation; phase; oscillations of a spring-restoring force
and force constant; energy in S.H.M. Kinetic and potential energies; simple pendulum derivation
of expression for its time period.
Free, forced and damped oscillations (qualitative ideas only), resonance.
4
Chapter–15: Waves
Wave motion: Transverse and longitudinal waves, speed of wave motion, displacement relation
for a progressive wave, principle of superposition of waves, reflection of waves, standing waves
in strings and organ pipes, fundamental mode and harmonics, Beats, Doppler effect.
PRACTICALS Total Periods: 60
The record, to be submitted by the students, at the time of their annual examination, has to include:
Record of at least 15 Experiments [with a minimum of 6 from each section], to be performed by the
students.
Record of at least 5 Activities [with a minimum of 2 each from section A and section B], to be
demonstrated by the teachers.
Report of the project to be carried out by the students.
EVALUATION SCHEME
Time Allowed: Three hours Max. Marks: 30
Two experiments one from each section 8+8 Marks
Practical record (experiment and activities) 6 Marks
Investigatory Project 3 Marks
Viva on experiments, activities and project 5 Marks
Total 30 Marks
SECTION–A
Experiments
1. To measure diameter of a small spherical/cylindrical body and to measure internal diameter and depth
of a given beaker/calorimeter using Vernier Callipers and hence find its volume.
2. To measure diameter of a given wire and thickness of a given sheet using screw gauge.
3. To determine volume of an irregular lamina using screw gauge.
4. To determine radius of curvature of a given spherical surface by a spherometer.
5. To determine the mass of two different objects using a beam balance.
6. To find the weight of a given body using parallelogram law of vectors.
7. Using a simple pendulum, plot its L-T2 graph and use it to find the effective length of second's
pendulum.
8. To study variation of time period of a simple pendulum of a given length by taking bobs of same size but
different masses and interpret the result.
9. To study the relationship between force of limiting friction and normal reaction and to find the co-
efficient of friction between a block and a horizontal surface.
10. To find the downward force, along an inclined plane, acting on a roller due to gravitational pull of the
earth and study its relationship with the angle of inclination θ by plotting graph between force and sinθ.
5
Activities
(for the purpose of demonstration only)
1. To make a paper scale of given least count, e.g., 0.2cm, 0.5 cm.
2. To determine mass of a given body using a metre scale by principle of moments.
3. To plot a graph for a given set of data, with proper choice of scales and error bars.
4. To measure the force of limiting friction for rolling of a roller on a horizontal plane.
5. To study the variation in range of a projectile with angle of projection.
6. To study the conservation of energy of a ball rolling down on an inclined plane (using a double inclined
plane).
7. To study dissipation of energy of a simple pendulum by plotting a graph between square of amplitude
and time.
SECTION–B
Experiments
1. To determine Young's modulus of elasticity of the material of a given wire.
2. To find the force constant of a helical spring by plotting a graph between load and extension.
3. To study the variation in volume with pressure for a sample of air at constant temperature by plotting
graphs between P and V, and between P and 1/V.
4. To determine the surface tension of water by capillary rise method.
5. To determine the coefficient of viscosity of a given viscous liquid by measuring terminal velocity of a
given spherical body.
6. To study the relationship between the temperature of a hot body and time by plotting a cooling curve.
7. To determine specific heat capacity of a given solid by method of mixtures.
8. To study the relation between frequency and length of a given wire under constant tension using
sonometer.
9. To study the relation between the length of a given wire and tension for constant frequency using
sonometer.
10. To find the speed of sound in air at room temperature using a resonance tube by two resonance
positions.
Activities
(for the purpose of demonstration only)
1. To observe change of state and plot a cooling curve for molten wax.
2. To observe and explain the effect of heating on a bi-metallic strip.
3. To note the change in level of liquid in a container on heating and interpret the observations.
4. To study the effect of detergent on surface tension of water by observing capillary rise.
5. To study the factors affecting the rate of loss of heat of a liquid.
6. To study the effect of load on depression of a suitably clamped metre scale loaded at (i) its end (ii) in
the middle.
7. To observe the decrease in presure with increase in velocity of a fluid.
6
Practical Examination for Visually Impaired Students
Class XI
Note: Same Evaluation scheme and general guidelines for visually impaired students as given for Class XII
may be followed.
A. Items for Identification/Familiarity of the apparatus for assessment in practicals(All experiments)
Spherical ball, Cylindrical objects, vernier calipers, beaker, calorimeter, Screw gauge, wire, Beam
balance, spring balance, weight box, gram and milligram weights, forceps, Parallelogram law of vectors
apparatus, pulleys and pans used in the same ‘weights’ used, Bob and string used in a simple
pendulum, meter scale, split cork, suspension arrangement, stop clock/stop watch, Helical spring,
suspension arrangement used, weights, arrangement used for measuring extension, Sonometer,
Wedges, pan and pulley used in it, ‘weights’ Tuning Fork, Meter scale, Beam balance, Weight box, gram
and milligram weights, forceps, Resonance Tube, Tuning Fork, Meter scale, Flask/Beaker used for
adding water.
B. List of Practicals
1. To measure diameter of a small spherical/cylindrical body using vernier calipers.
2. To measure the internal diameter and depth of a given beaker/calorimeter using vernier calipers
and hence find its volume.
3. To measure diameter of given wire using screw gauge.
4. To measure thickness of a given sheet using screw gauge.
5. To determine the mass of a given object using a beam balance.
6. To find the weight of given body using the parallelogram law of vectors.
7. Using a simple pendulum plot L-T and L-T2 graphs. Hence find the effective length of second’s
pendulum using appropriate length values.
8. To find the force constant of given helical spring by plotting a graph between load and extension.
9. (i) To study the relation between frequency and length of a given wire under constant tension
using a sonometer.
(ii) To study the relation between the length of a given wire and tension, for constant frequency,
using a sonometer.
10. To find the speed of sound in air, at room temperature, using a resonance tube, by observing the
two resonance positions.
Note: The above practicals may be carried out in an experiential manner rather than recording observations.
Prescribed Books:
1. Physics Part-I, Textbook for Class XI, Published by NCERT
2. Physics Part-II, Textbook for Class XI, Published by NCERT
3. The list of other related books and manuals brought out by NCERT (consider multimedia also).
7
PHYSICS (Code No. 042)
QUESTION PAPER DESIGN
CLASS - XI (2015-16)
Time 3 Hours Max. Marks: 70
S.
No.
Typology of Questions Very Short
Answer
(VSA)
(1 mark)
Short
Answer-I
(SA-I)
(2 marks)
Short
Answer –II
(SA-II)
(3 marks)
Value
based
question
(4 marks)
Long
Answer
(LA)
(5 marks)
Total
Marks
%
Weightage
1. Remembering- (Knowledge
based Simple recall questions,
to know specific facts, terms,
concepts, principles, or
theories, identify, define, or
recite information)
2 1 1 – – 7 10%
2 Understanding-
(Comprehension -to be
familiar with meaning and to
understand conceptually,
interpret, compare, contrast,
explain, paraphrase
information)
– 2 4 – 1 21 30%
3 Application - (Use abstract -
information in concrete
situation, to apply knowledge
to new situations, Use given
content to interpret a
situation, provide an example,
or solve a problem)
– 2 4 – 1 21 30%
4 High Order Thinking Skills -
(Analysis & Synthesis- Classify,
compare, contrast, or
differentiate between
different pieces of
information, Organize and/or
integrate unique pieces of
information from a variety of
sources)
2 – 1 – 1 10 14%
5 Evaluation - (Appraise, judge,
and/or justify the value or
worth of a decision or
outcome, or to predict
outcomes based on values)
1 – 2 1 – 11 16%
TOTAL 5x1=5 5x2=10 12x3=36 1x4=4 3x5=15 70(26) 100%
8
Question Wise Break Up
Type of Question Mark per Question Total No. of Questions Total Marks
VSA 1 5 05
SA-I 2 5 10
SA-II 3 12 36
VBQ 4 1 04
LA 5 3 15
Total 26 70
1. Internal Choice: There is no overall choice in the paper. However, there is an internal choice in one
question of 2 marks weightage, one question of 3 marks weightage and all the three questions of 5
marks weightage.
2. The above template is only a sample. Suitable internal variations may be made for generating
similar templates keeping the overall weightage to different form of questions and typology of
questions same.
9
CLASS XII (2015-16)
(THEORY)
Time: 3 hrs. Max Marks: 70
No. of Periods Marks
Unit–I Electrostatics 22
15
Chapter–1: Electric Charges and Fields
Chapter–2: Electrostatic Potential and Capacitance
Unit-II Current Electricity 20
Chapter–3: Current Electricity
Unit-III Magnetic Effects of Current and Magnetism 22
16
Chapter–4: Moving Charges and Magnetism
Chapter–5: Magnetism and Matter
Unit-IV Electromagnetic Induction and Alternating Currents 20
Chapter–6: Electromagnetic Induction
Chapter–7: Alternating Current
Unit–V Electromagnetic Waves 04
17
Chapter–8: Electromagnetic Waves
Unit–VI Optics 25
Chapter–9: Ray Optics and Optical Instruments
Chapter–10: Wave Optics
Unit–VII Dual Nature of Radiation and Matter 08
10
Chapter–11: Dual Nature of Radiation and Matter
Unit–VIII Atoms and Nuclei 14
Chapter–12: Atoms
Chapter–13: Nuclei
Unit–IX Electronic Devices 15
12
Chapter–14: Semiconductor Electronics: Materials,
Devices and Simple Circuits
Unit–X Communication Systems 10
Chapter–15: Communication Systems
Total 160 70
10
Unit I: Electrostatics 22 Periods
Chapter–1: Electric Charges and Fields
Electric Charges; Conservation of charge, Coulomb's law-force between two point charges,
forces between multiple charges; superposition principle and continuous charge distribution.
Electric field, electric field due to a point charge, electric field lines, electric dipole, electric
field due to a dipole, torque on a dipole in uniform electric fleld.
Electric flux, statement of Gauss's theorem and its applications to find field due to infinitely
long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical
shell (field inside and outside).
Chapter–2: Electrostatic Potential and Capacitance
Electric potential, potential difference, electric potential due to a point charge, a dipole and
system of charges; equipotential surfaces, electrical potential energy of a system of two point
charges and of electric dipole in an electrostatic field.
Conductors and insulators, free charges and bound charges inside a conductor. Dielectrics and
electric polarisation, capacitors and capacitance, combination of capacitors in series and in
parallel, capacitance of a parallel plate capacitor with and without dielectric medium between
the plates, energy stored in a capacitor.
Unit II: Current Electricity 20 Periods
Chapter–3: Current Electricity
Electric current, flow of electric charges in a metallic conductor, drift velocity, mobility and
their relation with electric current; Ohm's law, electrical resistance, V-I characteristics (linear
and non-linear), electrical energy and power, electrical resistivity and conductivity, Carbon
resistors, colour code for carbon resistors; series and parallel combinations of resistors;
temperature dependence of resistance.
Internal resistance of a cell, potential difference and emf of a cell, combination of cells in
series and in parallel, Kirchhoff's laws and simple applications, Wheatstone bridge, metre
bridge.
Potentiometer - principle and its applications to measure potential difference and for
comparing EMF of two cells; measurement of internal resistance of a cell.
Unit III: Magnetic Effects of Current and Magnetism 22 Periods
Chapter–4: Moving Charges and Magetism
Concept of magnetic field, Oersted's experiment.
Biot - Savart law and its application to current carrying circular loop.
Ampere's law and its applications to infinitely long straight wire. Straight and toroidal solenoids
(only qualitative treatment), force on a moving charge in uniform magnetic and electric fields,
Cyclotron.
Force on a current-carrying conductor in a uniform magnetic field, force between two parallel
current-carrying conductors-definition of ampere, torque experienced by a current loop in
uniform magnetic field; moving coil galvanometer-its current sensitivity and conversion to
ammeter and voltmeter.
11
Chapter–5: Magnetism and Matter
Current loop as a magnetic dipole and its magnetic dipole moment, magnetic dipole moment of
a revolving electron, magnetic field intensity due to a magnetic dipole (bar magnet) along its
axis and perpendicular to its axis, torque on a magnetic dipole (bar magnet) in a uniform
magnetic field; bar magnet as an equivalent solenoid, magnetic field lines; earth's magnetic
field and magnetic elements.
Para-, dia- and ferro - magnetic substances, with examples. Electromagnets and factors
affecting their strengths, permanent magnets.
Unit IV: Electromagnetic Induction and Alternating Currents 20 Periods
Chapter–6: Electromagnetic Induction
Electromagnetic induction; Faraday's laws, induced EMF and current; Lenz's Law, Eddy currents.
Self and mutual induction.
Chapter–7: Alternating Current
Alternating currents, peak and RMS value of alternating current/voltage; reactance and
impedance; LC oscillations (qualitative treatment only), LCR series circuit, resonance; power in
AC circuits, wattless current.
AC generator and transformer.
Unit V: Electromagnetic waves 04 Periods
Chapter–8: Electromagnetic Waves
Basic idea of displacement current, Electromagnetic waves, their characteristics, their
Transverse nature (qualitative ideas only).
Electromagnetic spectrum (radio waves, microwaves, infrared, visible, ultraviolet, X-rays,
gamma rays) including elementary facts about their uses.
Unit VI: Optics 25 Periods
Chapter–9: Ray Optics and Optical Instruments
Ray Optics: Reflection of light, spherical mirrors, mirror formula, refraction of light, total
internal reflection and its applications, optical fibres, refraction at spherical surfaces, lenses,
thin lens formula, lensmaker's formula, magnification, power of a lens, combination of thin
lenses in contact, combination of a lens and a mirror, refraction and dispersion of light through
a prism.
Scattering of light - blue colour of sky and reddish apprearance of the sun at sunrise and sunset.
Optical instruments: Microscopes and astronomical telescopes (reflecting and refracting) and
their magnifying powers.
Chapter–10: Wave Optics
Wave optics: Wave front and Huygen's principle, reflection and refraction of plane wave at a
plane surface using wave fronts. Proof of laws of reflection and refraction using Huygen's
principle. Interference, Young's double slit experiment and expression for fringe width,
coherent sources and sustained interference of light, diffraction due to a single slit, width of
central maximum, resolving power of microscope and astronomical telescope, polarisation,
plane polarised light, Brewster's law, uses of plane polarised light and Polaroids.
12
Unit VII: Dual Nature of Radiation and Matter 08 Periods
Chapter–11: Dual Nature of Radiation and Matter
Dual nature of radiation, Photoelectric effect, Hertz and Lenard's observations; Einstein's
photoelectric equation-particle nature of light.
Matter waves-wave nature of particles, de-Broglie relation, Davisson-Germer experiment
(experimental details should be omitted; only conclusion should be explained).
Unit VIII: Atoms and Nuclei 14 Periods
Chapter–12: Atoms
Alpha-particle scattering experiment; Rutherford's model of atom; Bohr model, energy levels,
hydrogen spectrum.
Chapter–13: Nuclei
Composition and size of nucleus, Radioactivity, alpha, beta and gamma particles/rays and their
properties; radioactive decay law.
Mass-energy relation, mass defect; binding energy per nucleon and its variation with mass
number; nuclear fission, nuclear fusion.
Unit IX: Electronic Devices 15 Periods
Chapter–14: Semiconductor Electronics: Materials, Devices and Simple Circuits
Energy bands in conductors, semiconductors and insulators (qualitative ideas only)
Semiconductor diode - I-V characteristics in forward and reverse bias, diode as a rectifier;
Special purpose p-n junction diodes: LED, photodiode, solar cell and Zener diode and their
characteristics, zener diode as a voltage regulator.
Junction transistor, transistor action, characteristics of a transistor and transistor as an
amplifier (common emitter configuration), basic idea of analog and digital signals, Logic gates
(OR, AND, NOT, NAND and NOR).
Unit X: Communication Systems 10 Periods
Chapter–15: Communication Systems
Elements of a communication system (block diagram only); bandwidth of signals (speech, TV and
digital data); bandwidth of transmission medium. Propagation of electromagnetic waves in the
atmosphere, sky and space wave propagation, satellite communication. Need for modulation,
amplitude modulation and frequency modulation, advantages of frequency modulation over
amplitude modulation. Basic ideas about internet, mobile telephony and global positioning
system (GPS)
PRACTICALS (Total Periods 60)
The record to be submitted by the students at the time of their annual examination has to include:
Record of at least 15 Experiments [with a minimum of 6 from each section], to be performed by the
students.
Record of at least 5 Activities [with a minimum of 2 each from section A and section B], to be
demonstrated by the teachers.
The Report of the project to be carried out by the students.
13
Evaluation Scheme
Time Allowed: Three hours Max. Marks: 30
Two experiments one from each section 8+8 Marks
Practical record [experiments and activities] 6 Marks
Investigatory Project 3 Marks
Viva on experiments, activities and project 5 Marks
Total 30 marks
SECTION–A
Experiments
1. To determine resistance per cm of a given wire by plotting a graph for potential difference versus
current.
2. To find resistance of a given wire using metre bridge and hence determine the resistivity (specific
resistance) of its material.
3. To verify the laws of combination (series) of resistances using a metre bridge.
4. To verify the laws of combination (parallel) of resistances using a metre bridge.
5. To compare the EMF of two given primary cells using potentiometer.
6. To determine the internal resistance of given primary cell using potentiometer.
7. To determine resistance of a galvanometer by half-deflection method and to find its figure of merit.
8. To convert the given galvanometer (of known resistance and figure of merit) into a voltmeter of desired
range and to verify the same.
9. To convert the given galvanometer (of known resistance and figure of merit) into an ammeter of desired
range and to verify the same.
10. To find the frequency of AC mains with a sonometer.
Activities
(For the purpose of demonstration only)
1. To measure the resistance and impedance of an inductor with or without iron core.
2. To measure resistance, voltage (AC/DC), current (AC) and check continuity of a given circuit using
multimeter.
3. To assemble a household circuit comprising three bulbs, three (on/off) switches, a fuse and a power
source.
4. To assemble the components of a given electrical circuit.
5. To study the variation in potential drop with length of a wire for a steady current.
6. To draw the diagram of a given open circuit comprising at least a battery, resistor/rheostat, key,
ammeter and voltmeter. Mark the components that are not connected in proper order and correct the
circuit and also the circuit diagram.
SECTION–B
Experiments
1. To find the value of v for different values of u in case of a concave mirror and to find the focal length.
14
2. To find the focal length of a convex mirror, using a convex lens.
3. To find the focal length of a convex lens by plotting graphs between u and v or between 1/u and 1/v.
4. To find the focal length of a concave lens, using a convex lens.
5. To determine angle of minimum deviation for a given prism by plotting a graph between angle of
incidence and angle of deviation.
6. To determine refractive index of a glass slab using a travelling microscope.
7. To find refractive index of a liquid by using convex lens and plane mirror.
8. To draw the I-V characteristic curve for a p-n junction in forward bias and reverse bias.
9. To draw the characteristic curve of a zener diode and to determine its reverse break down voltage.
10. To study the characteristic of a common - emitter npn or pnp transistor and to find out the values of
current and voltage gains.
Activities
(For the purpose of demonstration only)
1. To identify a diode, an LED, a transistor, an IC, a resistor and a capacitor from a mixed collection of
such items.
2. Use of multimeter to (i) identify base of transistor, (ii) distinguish between npn and pnp type transistors,
(iii) see the unidirectional flow of current in case of a diode and an LED, (iv) check whether a given
electronic component (e.g., diode, transistor or IC) is in working order.
3. To study effect of intensity of light (by varying distance of the source) on an LDR.
4. To observe refraction and lateral deviation of a beam of light incident obliquely on a glass slab.
5. To observe polarization of light using two Polaroids.
6. To observe diffraction of light due to a thin slit.
7. To study the nature and size of the image formed by a (i) convex lens, (ii) concave mirror, on a screen
by using a candle and a screen (for different distances of the candle from the lens/mirror).
8. To obtain a lens combination with the specified focal length by using two lenses from the given set of
lenses.
Suggested Investigatory Projects
1. To study various factors on which the internal resistance/EMF of a cell depends.
2. To study the variations in current flowing in a circuit containing an LDR because of a variation in
(a) the power of the incandescent lamp, used to 'illuminate' the LDR (keeping all the lamps at a fixed
distance).
(b) the distance of a incandescent lamp (of fixed power) used to 'illuminate' the LDR.
3. To find the refractive indices of (a) water (b) oil (transparent) using a plane mirror, an equi convex lens
(made from a glass of known refractive index) and an adjustable object needle.
4. To design an appropriate logic gate combination for a given truth table.
5. To investigate the relation between the ratio of (i) output and input voltage and (ii) number of turns in
the secondary coil and primary coil of a self designed transformer.
6. To investigate the dependence of the angle of deviation on the angle of incidence using a hollow prism
filled one by one, with different transparent fluids.
15
7. To estimate the charge induced on each one of the two identical styrofoam (or pith) balls suspended in a
vertical plane by making use of Coulomb's law.
8. To set up a common base transistor circuit and to study its input and output characteristic and to
calculate its current gain.
9. To study the factor on which the self inductance of a coil depends by observing the effect of this coil,
when put in series with a resistor/(bulb) in a circuit fed up by an A.C. source of adjustable frequency.
10. To construct a switch using a transistor and to draw the graph between the input and output voltage and
mark the cut-off, saturation and active regions.
11. To study the earth's magnetic field using a tangent galvanometer.
Practical Examination for Visually Impaired Students of Classes XI and XII
Evaluation Scheme
Time Allowed: Two hours Max. Marks: 30
Identification/Familiarity with the apparatus 5 marks
Written test (based on given/prescribed practicals) 10 marks
Practical Record 5 marks
Viva 10 marks
Total 30 marks
General Guidelines
The practical examination will be of two hour duration.
A separate list of ten experiments is included here.
The written examination in practicals for these students will be conducted at the time of practical
examination of all other students.
The written test will be of 30 minutes duration.
The question paper given to the students should be legibly typed. It should contain a total of 15
practical skill based very short answer type questions. A student would be required to answer any 10
questions.
A writer may be allowed to such students as per CBSE examination rules.
All questions included in the question papers should be related to the listed practicals. Every question
should require about two minutes to be answered.
These students are also required to maintain a practical file. A student is expected to record at least
five of the listed experiments as per the specific instructions for each subject. These practicals should
be duly checked and signed by the internal examiner.
The format of writing any experiment in the practical file should include aim, apparatus required,
simple theory, procedure, related practical skills, precautions etc.
Questions may be generated jointly by the external/internal examiners and used for assessment.
The viva questions may include questions based on basic theory/principle/concept, apparatus/
materials/chemicals required, procedure, precautions, sources of error etc.
16
Class XII
A. Items for Identification/ familiarity with the apparatus for assessment in practicals (All experiments)
Meter scale, general shape of the voltmeter/ammeter, battery/power supply, connecting wires,
standard resistances, connecting wires, voltmeter/ammeter, meter bridge, screw gauge, jockey
Galvanometer, Resistance Box, standard Resistance, connecting wires, Potentiometer, jockey,
Galvanometer, Lechlanche cell, Daniell cell (simple distinction between the two vis-à-vis their outer
(glass and copper) containers, rheostat connecting wires, Galvanometer, resistance box, Plug-in and
tapping keys, connecting wires battery/power supply, Diode, Transistor, IC, Resistor (Wire-wound or
carbon ones with two wires connected to two ends), capacitors (one or two types), Inductors, Simple
electric/electronic bell, battery/power supply, Plug-in and tapping keys, Convex lens, concave lens,
convex mirror, concave mirror, Core/hollow wooden cylinder, insulated wire, ferromagnetic rod,
Transformer core, insulated wire.
B. List of Practicals
1. To determine the resistance per cm of a given wire by plotting a graph between voltage and
current.
2. To verify the laws of combination (series/parallel combination) of resistances by ohm’s law.
3. To find the resistance of a given wire using a meter bridge and hence determine the specific
resistance (resistivity) of its material.
4. To compare the e.m.f of two given primary cells using a potentiometer.
5. To determine the resistance of a galvanometer by half deflection method.
6. To identify a
(i) diode, transistor and IC
(ii) resistor, capacitor and inductor, from a mixed collection of such items.
7. To understand the principle of (i) a NOT gate (ii) an OR gate (iii)an AND gate and to make their
equivalent circuits using a bell and cells/battery and keys /switches.
8. To observe the difference between
(i) a convex lens and a concave lens
(ii) a convex mirror and a concave mirror and to estimate the likely difference between the power
of two given convex /concave lenses.
9. To design an inductor coil and to know the effect of
(i) change in the number of turns
(ii) introduction of ferromagnetic material as its core material on the inductance of the coil.
10. To design a (i) step up (ii) step down transformer on a given core and know the relation between its
input and output voltages.
Note: The above practicals may be carried out in an experiential manner rather than recording observations.
Prescribed Books:
1. Physics, Class XI, Part -I and II, Published by NCERT.
2. Physics, Class XII, Part -I and II, Published by NCERT.
3. The list of other related books and manuals brought out by NCERT (consider multimedia also).
17
PHYSICS (Code No. 042)
QUESTION PAPER DESIGN
CLASS - XII (2015-16)
Time 3 Hours Max. Marks: 70
S.
No.
Typology of
Questions
Very Short
Answer
(VSA)
(1 mark)
Short
Answer-I
(SA-I)
(2 marks)
Short
Answer –II
(SA-II) (3
marks)
Value
based
question
(4 marks)
Long
Answer
(LA)
(5 marks)
Total
Marks
%
Weightage
1. Remembering - (Knowledge
based Simple recall
questions, to know
specific facts, terms,
concepts, principles, or
theories, Identify, define,
or recite, information)
2 1 1 – – 7 10%
2 Understanding -
(Comprehension -to be
familiar with meaning and
to understand conceptually,
interpret, compare,
contrast, explain,
paraphrase information)
– 2 4 – 1 21 30%
3 Application - (Use abstract
information in concrete
situation, to apply
knowledge to new
situations, Use given
content to interpret a
situation, provide an
example, or solve a
problem)
– 2 4 – 1 21 30%
4 High Order Thinking Skills -
(Analysis & Synthesis-
Classify, compare, contrast,
or differentiate between
different pieces of
information, Organize
and/or integrate unique
pieces of information from a
variety of sources)
2 – 1 – 1 10 14%
5 Evaluation - (Appraise,
judge, and/or justify the
value or worth of a decision
or outcome, or to predict
outcomes based on values)
1 – 2 1 – 11 16%
TOTAL 5x1=5 5x2=10 12x3=36 1x4=4 3x5=15 70(26) 100%
18
QUESTION WISE BREAK UP
Type of Question Mark per Question Total No. of Questions Total Marks
VSA 1 5 05
SA-I 2 5 10
SA-II 3 12 36
VBQ 4 1 04
LA 5 3 15
Total 26 70
1. Internal Choice: There is no overall choice in the paper. However, there is an internal choice in one
question of 2 marks weightage, one question of 3 marks weightage and all the three questions of 5
marks weightage.
2. The above template is only a sample. Suitable internal variations may be made for generating
similar templates keeping the overall weightage to different form of questions and typology of
questions same.
19
20
DELHI STATE
SCIENCE
TEACHERS FORUM
21
List of Experts who contributed in this analysis under the aegis of DSSTF
1. Mr P.C.Bose Retd. D. D of Education
2. Mr. P.N. Varshney Retd. Principal and President DSSTF
3. Dr. R.A. Goyal Retd Principal
4. Mr. Kanhiya Lal Retd Principal
5. Mr. A.P.Agarwal Retd Principal
6. Dr. M.S.Bhandari PGT Physics, Happy School
7. Mr. R.Rangarajan PGT Physics, DTEA School, Lodhi Road.
8. Mr. Arunachalam PGT Physics, Sumermal Jain Public School, Janapuri
9. Mrs. Girija Shankar Retd. PGT Physics, Pratibha Vikas Vidyalaya
10. Mrs. A.B.T. Sundari PGT Physics, Andhra School, Prasaad Nagar
11. Mr. N.C.Jain Retd PGT Physics, No2, Luedlo Castle
12. Mrs. Archana Joshi PGT Physics, SBV, Rajouri Garden Main
13. Mr. Brijesh Kumar PGT Physics, Takshila Public School
14. Mrs. Sujatha PGT Physics, S.B.V, Ramesh Nagar
15. Mrs. Jothi Singh PGT Physics, Lady Irwin Sr. Sec School
22
Question-wise analysis of CBSE Question Paper – 2015
CLASS- XII SUBJECT – PHYSICS Code No. 55/1/1/D
Q.
No.
Unit Form of
Question
Difficult
Level
E/AV/D/H
Learning
Outcome
Quality Of
Question
Translation Remark
1. Electrostatics Very Short Easy Knowledge Needs
improvement
The word
capacitive instead
of Capacitor may
be more
appropriate
2. Electrostatics Very Short Easy Understanding Precise
3. Optics Very Short Average Application Ambiguous In Hindi lens
given is
Convex,
while in
English it is
Concave
lens.
Translation
is wrong.
4. Communication Very Short Average Knowledge Highly
ambiguous
5. Current
Electricity
Very Short Average Understanding Diagram is
not Precise
for the II
portion of
the
question.(sho
uld be a
straight line)
Half mark for the
II portion should
be awarded for II
portion
6. Dual Nature Short
Answer –I
Above
Average
Application Precise
7. Atoms Short
Answer – I
Easy Knowledge Precise
8. Solids and
Semiconductors
Short
Answer – I
Easy Knowledge Precise Needs
improvement The number of
differences
expected should
have been
specified in the
question. For 2
differences
marks should be
awarded
9. Optics Short
Answer – I
Difficult Application Precise Inequalities is not
specified in the
syllabus
23
OR
part
Difficult Application Ambiguous Marks are to be
awarded whether
the student takes
the intensity of
polarized or the
un-polarised
light.
Even with un-
simplified
expression, 1.5
marks are to be
awarded.
10. Current
Electricity
Short
Answer – I
Average Knowledge Precise.
11. E.M waves Short
Answer – II
Easy Knowledge Precise. Production of
E.M.Waves is not
in syllabus, so full
marks should be
given for the
identification of
waves.
12. Optics Short
Answer – II
Difficult Application Precise. If the student
solve the
I part for the
magnification at
near point marks
should be given.
13. Dual Nature of
Matter
Short
Answer – II
Difficult
Knowledge
and
Understanding
Precise.
Lengthy for a 3
mark question.
[2+1]
14. Nucleus Short
Answer – II
Average Understanding Not a good
formulation
The framing of
question is not
appropriate.
Ambiguous
answers may evolve. [1+1+1]
OR
Part
Average Knowledge
and
Application
Too Lengthy In the I part
number of points
of difference must
be mentioned.
15. Communication Short
Answer – II
Average Skill Precise
16. Current
Electricity
Short
Answer – II
Difficult Skill Not a good
formulation Conditions are not
clear to draw the
graph. [ Eq 1 +
1+1]
17. Electrostatics Short
Answer – II
Difficult Knowledge Precise
Calculation is very
lengthy for a three
mark question. For
the two equations
24
1 mark each and
for the calculation
of charge1 mark.
No mark should
be cut for the
calculation part.
18. Magnetic
Effects of
Current
Short
Answer – II
Difficult Application Not a precise
format
Language in the
English format is
misguiding.
Wording in
English format is
not matching the
Hindi Translation
19. Solids and
Semiconductors
Short
Answer –II Easy Knowledge
and
Understanding
Precise
[1+1+1]
20. Solids and
Semiconductors
Short
Answer -II
Easy Understanding
and Skill
Precise [1+1+1]
21. Optics Short
Answer -II
Average Understanding
and
Application
Precise 11/2+11/2
22. Alternating
Current
Short
Answer –II
Average Application Precise 1+1+1
23. Alternating
Current
Value Based Easy Application Not Precise Language should
be “a.c” and not
“a.c. current”
24. Magnetism
Long
Answer
Average Knowledge
and
Application
Precise Too lengthy for 5
marks. II part is
beyond the
students
comprehension.
Marks are to be
awarded even if NIA is not proved.
OR
Part
Electro
Magnetic
Induction
Average Knowledge
and
Understanding
Not a Precise
format in the
3rd
part
III part is not clear
as orientation of
coils is not
specified.
25. Optics Long
Answer Average Knowledge &
Application Precise.
OR
Part
Long
Answer Average Knowledge
and
Application
Precise
26. Electroststics Long
Answer
Average Knowledge &
Application
Precise
OR
Part
Electroststics Difficult (b) part is
not precise
“x” is not shown
in the figure.
25
Unit Wise Distribution of Marks
Unit Name of the UnitMarks allotted Marks in the paper
I Electrostatics
II Current Electricity 15
III Magnetic effect of current & Magnetism
IV Electromagnetic Induction and Alternating current 16
V Electromagnetic Waves
VI Optics 17
VII Dual Nature of Matter
VIII Atoms and Nuclei 10
IX Electronic Devices
X Communication Systems 12
Total 70
26
Difficulty Level Wise Distribution of Marks
S.No. Estimated difficulty level Marks
1. Easy 27% [19/70]
2. Average 46% [32/70]
3. Difficult 27% [19/70]
*There is a shift of 5% from Average to Easy and Average to Difficult category. *Estimated difficulty level percentage is closely matching with the design of the question paper. *The questioning in the paper needs to undergo a thorough check for achieving higher standard. *Weight-age of numerical problems is closely matching with the design of the question paper.
*Students on a larger scale have not appreciated the paper. *Most of the questions are taken directly from a single book - NCERT.
Series : OSR/1 Code No. 55/1/1DPHYSICS (Theory)
[Time allowed : 3 hours] [Maximum marks : 70]
General Instructions:(i) All questions are compulsory.(ii) There are 26 questions in total. All questions are complusory.(iii) This question paper has five sections: Section A, Section B, Section C, Section D and Section E.(iv) Section A contains (question Nos. 1 to 5) are very short answer type questions and carry one
mark each.(v) Section B contains (question Nos. 6 to 10) carry two marks each. Section C contains (question
Nos. 11 to 22) carry three marks each and Section D contains value based question (questionno. 23) carry four marks each. Section E contains (questin no. 24 to 26) carry five marks each.
(vi) There is no overall choice. However, an internal choice has been provided in one question oftwo marks, one question of three marks and all three questions of five marks each weightage.You have to attempt only one of the choices in such questions.
(vii) Use of calculators is not permitted. However, you may use log tables if necessary.(viii) You may use the following values of physical constants wherever necessary :
c = 3 × 108 m/s h = 6.63 × 10–34 Js e = 1.6 × 10–19C µ = 4 × 10–7 T mA-1
0
14 = 9 × l09 Nm2 C–2
SECTION – A1. Define capacitor reactance. Write its S.I. units.2. What is the electric flux through a cube of side 1 cm which encloses an electric dipole?3. A concave lens of refractive index 1.5 is immersed in a medium of refractive index 1.65. What
is the nature of the lens?4. How are side bands produced?5. Graph showing the variation of current versus voltage for a material GaAs is shown in the
figure. Identify the region of(a) negative resistance.(b) where Ohm’s law is obeyed.
SECTION – B6. A proton and an -particle have the same de-Broglie wavelength. Determine the ratio of (i)
their accelerating potentials (b) their speeds.7. Show that the radius of the orbit in hydrogen atom varies as n2, where n is the principal
quantum number of atom.8. Distinguish between intrinsic and extrinsic semiconductors.9. Use the mirror equation to show that an object placed between f and 2f of a concave mirror
produces a real image beyond 2f.OR
Find an expression for intensity of transmitted light when a polaroid sheet is rotated betweentwo crossed polaroids. In which position of the polaroid sheet will the transmitted intensitybe maximum?
10. Use Kirchhoff’s rule to obtain conditions for the balance condition in a Wheatstone bridge.
A
B
C D
E
Current I
Voltage V
SECTION – C11. Name the parts of the electromagnetic spectrum which is
(a) suitable for radar systems used in aircraft navitation.(b) used to treat muscular strain.(c) used as a diagnostic tool in medicine.Write in brief, how these waves can be produced.
12. (a) A giant refracting telescope has an objective lens of focal length 15 m. If an eye piece offocal length 1.0 cm is used, what is the angular magnification of the telescope?
(b) If this telescope is used to view the moon, what is the diameter of the image the moonformed by the objective lens? The diameter of the moon is 3.48 × 106 m and the radius oflunar orbit is 3.8 × 108 m.
13. Write Einstein’s photoelectric equation and mention which important features in photoelectriceffect can be explained with the help of this equation.The maximum kinetic energy of the photoelectrons gets doubled when the wavelength oflight incident on the surface changes from 1 to 2. Derive the expressions for the thresholdwavelength 0 and work function for the metal surface.
14. In the study of Geiger-Marsden experiment on scattering of -particles by a thin foil of gold,draw the trajectory of -particles in the coulomb field of target nucleus. Explain briefly howone gets the information on the size of the nucleus from this study.From the relation R = R0 A
1/3, where R0 is constant and A is the mass number of the nucleus,show that nuclear matter density is independent of A.
ORDistinguish between nuclear fission and fusion. Show how in both these processes energy isreleased.Calculate the energy release in MeV in the deuterium-tritium fusion reaction:
42 31 1 2H H He n
Using the data:21m( H) = 2.014102 u 3
1m( H) = 3.016049 u 42m( He) = 4.002603 u
mn = 1.008665 u 1u = 931.5 MeV/c2
15. Draw a block diagram of a detector for AM signal and show, using necessary processes andthe waveforms, how the original message signal is detected from the input AM waves.
16. A cell of emf ‘’ and internal resistance ‘r’ is connected across a variable load resistor R. Drawthe plots of the terminal voltage V versus (i) R and (ii) the current I.It is found that when R = 4, the current is 1A and when R is increased to 9, the currentreduces to 0.5A. Find the values of the emf and internal resistance r.
17. Two capacitors of unknown capacitances C1 and C2 are connected first in series and then inparallel across a battery of 100 V. If the energy stored in the two combinations is 0.045 J and0.25 J respectively, determine the value of C1 and C2. Also calculate the charge on eachcapacitor in parallel combination.
18. State the principle of working of a galvanometer.A galvanometer of resistance G is converted into a voltmeter to measure upto V volts byconnecting a resistance R1 in series with the coil. If a resistance R2 is connected in series withit, then it can measure upto V/2 volts. Find the resistance, in terms of R1 and R2, required tobe connected to convert it into a voltmeter that can read upto 2V. Also find the resistance Gof the galvanometer in terms of R1 and R2.
19. With what considerations in view, a photodiode is fabricated? State its working with the helpof a suitable diagram.
Eventhough the current in the forward bias is known to be more than in the reverse bias, yetthe photodiode works in reverse bias. What is the reason?
20. Draw a circuit diagram of a transistor amplifer in CE configuration.Define the terms:(a) Input resistance and (b) Current amplification factor.How are these determined using typical input and output characteristics?
21. Answer the following questions:(a) In a double slit experiment using light of wavelength 600 nm, the angular width of the
fringe formed on a distant screen is 0.1°. Find the spacing between the two slits.(b) Light of wavelength 5000 Å propagating in air gets partly reflected from the surface of
water. How will the wavelengths and frequencies of the reflected and refracted light beaffected?
22. An inductor L of inductance XL is connected in series with a bulb B and an ac source. Howwould brightness of the bulb change when (a) number of turn in the inductor is reduced, (b)an iron rod is inserted in the inductor and (c) a capacitor of reactance XC = XL is inserted inseries in the circuit. Justify your answer in each case.
SECTION – D23. A group of students while coming from the school noticed a box marked “Danger H.T. 2200
V” at a substation in the main street. They did not understand the utility of a such a highvoltage, while they argued, the supply was only 220 V. They asked their teacher this questionthe next day. The teacher thought it to be an important question and therefore explained tothe whole class.Answer the following questions:(a) What device is used to bring the high voltage down to low voltage of a.c. current and
what is the principle of its working?(b) Is it possible to use this device for bringing down the high dc voltage to the low voltage?
Explain.(c) Write the values displayed by the students and the teacher.
SECTION – E24. (a) State Ampere’s circuital law. Use this law to obtain the expression for the magnetic field
inside an air cored toroid of average radius ‘r’ having ‘n’ turns per unit length andcarrying a steady current I.
(b) An observer to the left of a solenoid of N turns each of crosssection area ‘A’ observes that a steady current I in it flows in theclockwise direction. Depict the magnetic field lines due to thesolenoid specifying its polarity and show that its acts as a barmagnet of magnetic moment m = NIA.
OR(a) Define mutual inductance and write its S.I. units.(b) Derive an expression for the mutual inductance of two long co-axial solendoids of same
length wound one over the other.(c) In an experiment, two coils C1 and C2 are placed close to each other. Find out the
expression for the emf induced in the coil C1 due to a change in the current through thecoil C2.
29
25. (a) Using Huygen’s construction of secondary wavelets explain how a diffraction pattern isobtained on a screen due to a narrow slit on which a monochromatic beam of light isincident normally.
(b) Show that the angular width of the first diffraction fringe is half that of the central fringe.
(c) Explain why the maxima at 12
na
become weaker and weaker with increasing n.
OR(a) A point object ‘O’ is kept in a medium of refractive index n1 in front of a convex spherical
surface of radius of curvature R which separates the second medium of refractive indexn2 from the first one, as shown in the figure.Draw the ray diagram showing the image formation and deduce the relationship betweenthe object distance and the image distance in terms of n1, n2 and R.
u R
Cn2
n1
O
(b) When the image formed above acts as a virtual object for a concave spherical surfaceseparating the medium n2 from n1(n2 > n1), draw this ray diagram and write the similar(similar to (a)) relation. Hence obtain the expression for the lens maker’s formula.
26. (a) An electric dipole of dipole moment p consists of point charges +q and – q separatedby a distance 2a apart. Deduce the expression for the electric field E
due to the dipole
at a distance x from the centre of the dipole on its axial line in terms of the dipole
moment p . Hence show that in the limit 30, 2 / 4 .x a E p x
(b) Given the electric field in the region 2 ,E xi find the net electric flux through the cube
and the charge enclosed by it.
az
x
y
OR(a) Explain, using suitable diagrams, the difference in the behaviour of a (i) conductor and
(ii) dielectric in the presence of external electric field. Define the terms polariazation ofa dielectric and write its relation with susceptibility.
2Q C
Q 2QA
(b) A thin metallic spherical shell of radius R carries a charge Q on its
surface. A point charge 2Q
is placed at its centre C and another
charge +2Q is placed outside the shell at a distance x from thecentre as shown in the figure. Find (i) the force on the charge at thecentre of shell and at the point A, (ii) the electric flux through theshell.
**************
30
Series : OSR/1 Code No. 55/1/1 DPHYSICS (Theory)
Solution to CBSE Board Examination 2014-2015
SECTION – A1. Opposition offered by the capacitor to the flow of a.c. through it is called capacitive reactance.
It is denoted by XC.
C1 1XC 2 fC
Its S.I. unit is ohm.2. Net flux = 0
Because net charge enclosed by cube = 0.3. Since lens < surrounding
It behaves like converging lens.4. Side bands are produced by the method of amplitude modulation.
It produces two new frequencies (fc + fm) and (fc – fm) around original frequency (fc), which arecalled side band frequencies.
Upper side band frequency = USB = fc + fm
and Lower side band frequency = LSB = fc – fm.5. (a) DE [ Slope is negative.]
(b) BC [ V I]
SECTION – B6. De-Broglie wavelength is given by
2
h h hp mv mqV , where, V = accelerating potential; v = speed of particle
As, Charge on proton = qp
Charge on -particle = q = 2qp
and mass of proton = mp
mass of -particle = m = 4mp
(i) Given = p
2 2 p p p
h hm q V m q V
mqV = mpqpVp
4 2p p p
p p p p
V m qm qV m q m q
81
(ii) As, = p
p p
h hm v m v
4p p
p p
v mmv m m
41
A
B
C D
E
Current I
Voltage V
31
7. According to Bohr’s theory, a hydrogen atom consists of a nucleus with a positive charge eand a single electron of charge –e, which revolves around it in circular orbit of radius r.The electrostatic force of attraction between the nucleus and the electron is
2
2
keFr
e , m–+e
Nucleus
rTo keep electron in its orbit, the centripetal force on the electronmust be equal to the electrostatic attraction. Therefore,
2 2
2
mv ker r
2
2 kemvr
2
2
kermv
...(i)
Where, m = mass of electronv = speed in an orbit of radius r.
Bohr’s quantisation condition for angular momentum is
L = mvr = nh2
nhv
2 mr ...(ii)
On substituting (ii) in (i), we get
2
2
kernhm
2 mr
2
2 2
2 2 2
kermn h
4 m r
22 2 2
2 2
ker 4 m rmn h
2 2
2 2
n hr4 mke
r n2 where, n = principal quantum number.
8. Intrinsic semi-conductors Extrinsic semi-conductors 1. These are pure semi-conducting
tetravalent crystals. 1. These are semi-conducting tetravalent
crystals doped with impurity atoms of group III or V.
2. Their electrical conductivity is low. 2. Their electrical conductivity is high. 3. There is no permitted energy state
between valence band and conduction bands
3. There is permitted energy state of the impurity atom between valence and conduction bands.
9. From mirror formula, 1 1 1v f u
Now, for a concave mirror, f < 0 and for an object on left, u < 0.
2f < u < f or 1 1 12f u f
or 1 1 12f u f or
1 1 1 1 1 1f 2f f u f f
or 1 1 02f v
32
This implies 1v
is negative or v is negative. So that image is formed on left. Also, the inequality
implies.2f > v or |2f| < |v| [ 2f and v are negative]
i.e. the real image is formed beyond 2f.OR
9. By Malus law, the intensity of light emerging from the middle polaroid C, will beI1 = I0 cos2 ; where I0 = intensity of light falling on middle polaroid.
Thus, intensity I1 falls on the polaroid at the end (polaroid B) whose polarisation axis makesan angle of (90° – ) with the polarisation axis of the angle of middle polaroid.Therefore, the intensity of light emerging from the polaroid B will be
I2 = I1 cos2 (90 – ) = (I0 cos2 ) cos2(90 – )
A BC
= I0 cos2 sin2 = 14 I0 (2 sin cos )2
202
II sin 24
Transmitted intensity I2 will be maximum when sin 2 = 1 or 2 = 90° or = 45°.
10. In accordance with Kirchoff’s first law, the currents through various branches are as shownin figure.
G
B
P
D
A C
Q
R S
(I – I )1 g
I2
Ig
I + I2 g
I
I1
I
Applyng Kirchoff’s second law to the loop ABDA, we getI1P + IgG – I2R = 0
where G is the resistance of the galvanometer. Again,applying Kirchoff’s second law to the loop BCDB, we get
(I1 – Ig)Q – (I2 + Ig)S – G Ig = 0In the balanced condition of the bridge Ig = 0. The aboveequations become
I1P – I2R = 0 or I1P = I2R ... (1)and I1Q – I2S = 0 or I1Q = I2S ... (2)On dividing equation (1) by (2), we get
P R=Q S
This proves the conditon for the balanced wheatstone bridge.
SECTION – C11. (a) Microwave
These waves can be produced by klystron valve or magnetron valve.(b) Infrared
These waves are produced by vibrations of atoms and molecules.(c) -rays
These waves are produced by the radioactive decay of the nucleus.
12. (a)
02
e
f 15|m| 1500f 1 10
(b) D = diameter of moon
D
r f0x
Objective lens
r = radius of lunar orbit
33
f0 = focal length of objectivex = diameter of image of moon
tan =
6
8
x 3.48 1015 3.8 10
6
8
3.48 10 15x3.8 10
= 13.73 × 10–2 m
x = 13.73 cm
13. Einstein’s photoelectric equation Photon½ m2
max
Photoelectron
Metal
h = 0 + Kmax
h = Energy of the photon0 = Work function of the metalKmax = Maximum kinetic energy of the emitted photoectron
2max max 0
1K mv h2
or Kmax = h – h0
whee, 0 = threshold frequency of metal surface.Explanation of features of photoelectric effect.(a) Einstein said that one photoelectron is ejected from a metal surface if one photon of
suitable light radiation falls on it. If the intensity of the light is increased, the number ofincident photon increases, which results in an increase in the number of photo-electronsejected. This implies photo current is proportional to intensity and radiation.
(b) If < 0, than maximum K.E. is negative, which is impossible. Hence photoelectric emissiondoes not take place for the incident radiation below threshold frequency.
(c) If > 0, maximum K.E. increases. This means, maximum K.E. of photoelectrons dependsonly the frequency of incident light.
Einstein equation corresponding to wavelength 1, max 0
1
hcK ...(i)
Einstein equation corresponding to wavelength 2, max 0
2
hc2K
0max
2
hcK2 2 ...(ii)
From (i) and (ii), we get
0
01 2
hc hc2 2 or
0
01 2
hc hc2 2
0
1 2
1 1hc2 2
01 2
1 12hc2
0
1 2
2 1hc
0 1 2
hc 2 1hc
1 20
2 12
34
14.
b Nucleus
r0
Suppose, an -particle with initial K.E. = 21 mv2 is directed towards the center of the
nuclues of an atom. On account of Coulomb’s repulsive force between nucleus and -particle, at the distance
of closest approach (r0) the particle stops and it cannot go closer to the nucleus and itsK.E. gets converted in P.E., i.e.,
2
0 0
1 (2 )2 4
Ze emvr
02
0
(2 )142
Ze ermv
Radius of the nucleus must be approximately equal to the ‘r0’. Nuclear density
Volume of nucleus = 34 R3 1/3 3
04 (R A )3
30
4 R A3
Density of nuclear matter = mass of nucleus
Volume of nucleus
33 00
A 34 4 RR A3
m m where m is the average mass of nucleus
m and R0 are constant
3
0
34 R
m
Density of nuclear matter is same for all elements.OR
14.
Nuclear Fission Nuclear Fusion It is the phenomenon of breaking of heavy nucleus to form two or more lighter nuclei.
It is the phenomenon of fusing two or more lighter nuclei to form a single heavy nucleus.
The products remain radioactive. The products are generally non-radioactive.
35
In both the processes, a certain mass (m) disappears, which appears in the form of energyas per Einstein equation : E = (m)c2. Given equation is
432211H H He n
Total mass of the reactant
3211( H) ( H)rm m m = (2.014102 + 3.016049)u = 5.030151u
Total mass of the product
42( He)p nm m m = (4.002603 + 1.008665)u = 5.011268 µ
m = mr – mp = (5.030151 – 5.011268)um = 0.018883u
Energy ReleasedE = mc2 = 0.018883 × 931.5 MeV = 17.589514 MeV
15. Block diagram of a deterctor
Envelopedetector
Rectifier Outputm(t)AM wave
Time OutputRectified wave
AM input waveDetection is the process of recovering the modutating signal from the modulated carrierwave.The modulated signal of the form given in (a) is passed through a rectifier to produce theoutput as shown in (b). This envelop of signal (b) is the message signal.To obtain modulating signal m(t) the signal in passed through an envelop detector.
16. V
R(a)
V
I(b)
Current in circuit is e=+( )
IR r
(VA – VB) = V = Terminal voltage = IR
e= = e -+( )RV Ir
R r
e=+( )
IR r
e= Þ + = e+
44
I rr ... (1)
e=+
0.59 r 0.5r + 4.5 = ... (2)
r = 1; = 5 volt
A r
R
B
36
17.
C1 C2
100V
C1
C2
100V
-= ´+
51 2
1 2
0.9 10C C
FC C C1 + C2 = 0.5 × 10–4F
1 2
1 2
9C C
FC C
= m+ C1 + C2 = 50µF
C1C2 = 450
C2 = 1
450C
+ =11
450 50CC
C12 – 50 C1 + 450 = 0
± -=150 2500 1800
2C = ± =25 175 11.8
C1 = 11.8 µF, C2 = 38.2 µF18. Principle:
Moving coil galvanometer is based on the fact that when a current carrying loop or coil isplaced in the uniform magnetic field, it experiences a torque.
igG
R1
V
V = ig (G + R1) ...(i)
igG
R2
V2
g 2V i (G R )2 ...(ii)
igG
R2V
2V = ig (G + R) ...(iii)
From (i) and (ii)
g 2 g 12i (G R ) i (G R )
2G + 2R2 = G + R1
G = R1 – 2R2 ...(iv)From (i) and (iii)
g 1 g2i (G R ) i (G R)
2G + 2R1 = G + RG = R – 2R1 ...(v)
From (iv) and (v)R1 – 2R2 = R – 2R1
R = 3R1 – 2R2 ...(vi)
045.021
2121 2
V
cc
cc+ =2
1 21 ( ) 0.252
C C V
37
Using (vi) in (v)G = R – 2R1
G = 3R1 – 2R2 – 2R1
G = R1 – 2R2
19. A photodiode is used to observe the change in current with change in the light intensityunder reverse bias condition.In fabrication of photodiode, material chosen should have band gap ~1.5 eV or lower so thatsolar conversion efficiency is better. This is the reason to choose Si or GaAs material.Working: It is a p – n junction fabricated with a transparent window to allow light photons tofall on it. These photons generate electron hole pairs upon absorption. If the juction is reversebiased using an electrical circuit, these electron hole pair move in opposite directions so asto produce current in the circuit. This current is very small and is detected by themicroammeter placed in the circuit.
np A
R. B.
I
I ( A)
(m A)
V
I >I >I >I4 3 2 1
I1I2I3I4
(a) Photodiode (R.B.) (b) I-V character of photodiode for different illumination intensities
A photodiode is preferably operated in reverse bias condition. Consider an n-typesemiconductor. Its majority carrier (electron) density is much larger than the minority holedensity i.e., n>>p. When illuminated with light, both types of carries increase euqally innumber
n' = n + n ; p' = p + pAs n >> p and n = p
That is, the fractional increase in majority carries is much less than the fractional increase inminority carriers. Consequently, the fractional change due to the photo-effects on the minoritycarrier dominated reverse bias current is more easily measurable than the fractional changein the majority carrier dominated forward bias current. Hence, photodiodes are preferableused in the reverse bias condition for measuring light intensity.
20. Transistor amplifer in CE configuration
IC
IB
RB
Vi IE
RC V0
VBB VCC
C
E
C
B
(a) Input resistance = Change in base-emitter voltage
Base current
pp
nn
38
ri = dynamic resistanceFrom the input characteristics we can calculate the change in VBE (VBE) and change in IB
(IB).(b) Current amplification factor ()
B
cdc
B
cac I
IandII
From the output characteristics we can calculate the change in IC (IC) and change in IB(IB).
20406080
100
0 0.2 0.4 0.6 0.8
IB
VCE = 10V
VBE
( A)
(V)
Input characteristics
2468
10IC
VCE
(mA)
(Volts)
Output characteristics50 A
30 A40 A
20 AI = 10 AB
21. (a) = 600 nm = 600 × 10–9 m = 0.1°
d or =
1.0180
10 600 –9
m
= 34394.90 × 10–8 = 0.343 × 10–3 m
(b) = 5000Å
n =lC
= 8 8
10 7
3 10 3 105000 10 5 10- -
´ ´=´ ´
= ´ 153 105 = ´ 1430 10
5 = 6 × 1014 Hz.
Frequency of reflected and refracted light is 6 × 1014HzVelocity of light in water
speed of light in airμ = speed of light in water
Therefore =
86
14
2.25 10' 0.375 10 m6 10
= 0.375 × 10–6 m
Wavelength of the refracted light is 0.375 × 10–6m
22. (i) = m2
0NL Al
If N is reduced L will decreaseAs XL = LXL will decrease. Current is increased and brightness is increased.
(ii) When iron rod inserted in the inductor L will increaseXL will also increasecurrent will decrease
B
BE
IV
d
v
810334
smv /1025.24
3103 88
39
so brightness will decrease(iii) A capacitor of reactance XC = XL in series in the circuit, due to it circuit attains resonance
condition.Total impedance will decrease, current will increase and brightness will increase.
SECTION – D23. (a) A step down transformer is used to bring high voltage to low voltage. It’s working is
based on mutual indirection.(b) No, because its working is based on electromagnetic induction, which is associated with
varying magnetic flux, but in case of dc source, current will be constant, flux will beconstant. This means we can not get output from transformer.
(c) Values displayed by students. Active. Student has investigative skills.Values displaced by teacher Patient, Motivating, ability to make use of subject knowledge of explain practical
application. Make use of modern technology.
SECTION – E
24. (a) The line integral of magnetic field B
around any closed path in vacuum is 0 times thetotal current through the closed path, i.e.
B dl
= 0I
Consider a circle of radius r.
Now,
. cosB dl Bdl
Angle between B and
dl is 0. Hence,
. cosB dl Bdl Bdl B dl
= B × circumference of the circle of radius r
or
. 2B dl B r ...(i)
According to Ampere’s circuital law,
.B dl = 0 × net current enclosed by the circle of radius r
= 0 × tota number of turns × I = 0 (n × 2r) I ...(ii) Comparing equation (i) and (ii), we get
B × 2r = 0 (n × 2r) I or B = 0nIWhich is the magnetic field due to a toroid carrying current.
(b) Solenoid as a magnetic dipole
(b)
(c)
S N
S N S NS N S N S NS Nd(a)
i
Each turn of the solenoid has been replaced by a dipole. The magnetic moment of eachturn is I × A. Since there are N turns the total magnetic moment of the solenoid ism = NIA. As shown in figure (b), intermediate poles neutralize each other and we are left
40
with the poles at the ends. Hence, the solenoid behaves like a bar magnet with southpole on the left and north pole on the right.
OR24. (a) Coefficient of mutual induction (M) or mutual inductance of two coils is equal to the
e.m.f. induced in one coil when rate of change of current through the other coil is unity.The SI unit of M is henry.
(b) Mutual inductance of two long co-axial solenoids B1 and B2 – Magnetic fields created by each solenoid. I1 and I2 – Current through each solenoid. 1 and 2 – Flux associated with each solenoid. N1 and N2 – Number of turns in each coil. l – Length of each solenoid.Pass current through S2, and record flux associated with S1
1 = N1B2A1
1 = (n1l) (0n2I2) (r12)
1 = M12I2
M12 = 0n1n2 r12 l = 0n1n2Al
Similarly pass current through S1 and record the flux associated with S2
2 = N2B1A1
2 = (n2 l) (0n1I1) (r12)
2 = M21I1
M21 = 0n1n2r12 l = 0n1n2Al
M21 = M12 = 0n1n2Al(c) I = strength of current in coil 2
Coil 2 Coil 1
= total amount of magnetic flux linked with all the turns of theneighbouring coil 1.It is found that
I or = MIwhere M is a constant of proportionality and is called coefficientof mutual induction or mutual inductance of the two coils.The e.m.f. induced in the neighbouring coil (1) is given by
= ddt
= d MIdt
= dIMdt
25. (a) Diffraction of light at a Single slitA single narrow slit is illuminated by a monochromatic source of light. The diffractionpattern is obtained on the screen placed infront of the slits.There is a central bright region called as central maximum. All the waves reaching thisregion are in phase hence the intensity is maximum.On both side of central maximum, there are alternate dark and bright regions, the intensitybecoming weaker away from the centre.The intensity at any point P on the screen depends on the path difference between thewaves arsing from different parts of the wavefront at the slit.The path difference (BP – AP) between the two edgesof the slit can be calculated using the figure.
Path difference, BP – AP = NQ= a sin a
At the central point C on the screen, the angle
41
is zero; therefore all path difference are zero andhence all the parts of slit contribute in phase.Due to this, the intensity at C is maximum.If this path difference is , (the wavelength of light used), then P will be point of minimumintensity. This is because the whole wvaefront can be considered to be divided into twoequal halves CA and CB and if the path difference between the secondary waves from Aand B is , then the path difference between the secondary waves from A and C reachingP will be /2, and path difference between the secondary waves from B and C reaching Pwill again be /2. Also, for every point in the upper half AC, there is a corresponing pointin the lower half CB for which the path difference between the secondary waves, reachingP is /2. Thus, destructive interference takes place at P and therefore, P is a point of firstsecondary minimum.
(b) Central bright lies between d
Angular width of central bright = 2 = l2
a ...(1)
first diffraction fringe lies between and lq = 2
a
Angular width of first difraction fringe is l l l- =2
a a a ...(2)
Hence proved from (1) and (2).(c) For the first maxima of diffraction pattern 2/3rd of the slit is responsible for destructive
interference. Hence first maxima is weaker than the central maxima. maxima gets weaker with increasing n.
OR25. (a) Refraction at spherical surface
Figure shows the geometry of formation of image I of an object O on the principal axis ofa spherical surface with centre of curvature C, and radius of curvature R.The rays are incident from a medium of refractive index n1, to another of refractive indexn2.
n1
u
MO
i
Rv
C
n2
N
r
IP
Approximation:(i) We take the aperture (or the lateral size) of the surface to be small compared to other
distances involved, so that small angle approximation can be made.(i) In particular NM will be taken to be nearly equal to the length of the perpenedicular
from the point N on the principal axis.For small angles, We have
C
P
A
a C
B
FromSource
N
42
MNtan NOM NOMOM
MNtan NCM NCMMC
MNtan NIM NIMMI
Now, for NOC, i is the exterior angle. Therefore, i =NOM + NCM
MN MNiOM MC
...(i)
Similarly, from NCIr = NMC – NIM
i.e.,MN MNrMC MI
...(ii)
Now, by Snell’s law n1 sin i = n2 sin rOr for small angles n1i = n2rSubstituting i and r from Equations, (i) and (ii), we get :
1 2MN MN MN MNn nOM MC MC MI
2 2 11 n n nnOM MI MC
...(iii)
Here, OM, MI and MC represent magnitudes of distances. Applying the Cartesian signconvention.OM = – u, MI = + v, MC = + R
Substituting these in equation (iii), we get : 2 2 11n n nnv u R
(b) n2 > n1
RC N P I
v
Mn1n2
ri
2
I1
v1
2 1 21
1 2Rn n nn
v v … (i)
from equation in part (a)
2 2 11
1 1Rn n nnv u … (ii)
Adding (i) and (ii)
1 1
2 11 2
1 1( )R R
n n n nv u
2
1 1 2
1 1 1 11R R
nv u n
u = , v = f
2
1 1 2
1 1 11R R
nf n
43
26. (a) Electric Field at a point on the axial line
2( )qkqE
x a
x–q +q
2ap qE qE
2( )qkqE
x a
2 2( ) ( )q qkq kqE E E
x a x a 2 2 24
( )kq axE
x a
2 2 22
( )k pxE
x a
(Parallel to p
)
If x >> a, then 30
24
pEx
or in vector form 3
0
24
pEx
(b) Since, the electric field is parallel to the faces parallel to xy and xz planes, the electricflux through them is zero.Electric flux through the left faceL = (EL) (a
2) cos 180° = (0) (a2) cos 180° = 0Electric flux through the right faceR = (ER) (a
2) cos 0° = (2a) (a2) × 1 = 2a3
Total flux () = 2a3 = 0
enclosedq making qenclosed = 2a3 0
OR26. (a) (i) Conductor
MetalEin
– +
E0
– +– +– +– +
E0 external fieldEin internal field created by the redistribution ofelectrons inside the metalWhen a conductor like a metal is subjected to externalelectric field, the electrons experience a force in theopposite direction collecting on the left hand side.A positive charge is therefore induced on the right hand side. This creates on oppositeelectric field (Ein) that balances out (E0) The net electric field inside the conductor becomes zero.
(ii) Dielectric
internal electric field
E0
– +
– +
– +
– +
When external electric field is applied, dipoles are created (in case of non-polardielectrics) or dipoles are aligned (in case of polar dielectrics). The placement ofdipoles is as shown in the given figure. An internal electric field is created whichreduces the external electric field.Polariazation of a dielectric (P) is defined as the dipole moment per unit volume ofthe polarized dielectric.
e 0P E Where e is susceptibility and E is Electric field
(b) (i) Since, the electric field inside a spherical shell is zero, the force on the chargeplaced at the centre of the shell is zero. For the charge at A, the shell will behave as
44
if the entire charge ‘Q’ is placed at the centre of the shell. Therefore, the total charge
is 3
2 2
Q QQ + = .
Since, its distance from 2Q is ‘x’, the electric field at A is 2
3
2
QK
Ex
=
So, electric force ( )2
2
0
1 32
4
QF Q x E x
xπε= =
(ii) Since, the total charge enclosed by the shell is 2
en
Qq = , the total flux according to
Gauss’s law is
0 0
2
2
φε ε
=
× · × · × · × · ×
CHANGES IN PHYSICS SYLLABUS
Following are some changes in the Syllabus of Physics at Senior Secondary Level by CBSE
Topic Class XI
1. Reynolds number removed
2. Newton’s law of cooling removed
3. Wein displacement law added
4. Stefan s law added
5. Green house effect added
Class XII
1. Vandegraff generator removed
2. Human eye and its defect removed
3. Combination of lens and mirror added
4. Atomic mass, isobars, isotops removed
5. Oscillator removed
6. Transistor as a switch removed
7. Basic idea about internet, mobile telephony and global positioning system added
45
Supplementary Material on Additional Sub-Topics
included in Class XI Physics Theory Syllabus
Radiation
• Radiation process does not need any material medium for heat transfer.
• Term Radiation refers to the continuous emission of energy from surface of all
bodies and this energy is called radiant energy.
• Radiant energy is in the form of Electro Magnetic waves.
• Radiant energy emitted by a sunface depends on the temperature and nature of the
surface.
• All bodies whether they are solid, liquid or gas emit radiant energy.
• EM radiations emitted by a body by virtue of increased temperature of a body are
called thermal radiation.
• Thermal radiation falling on a body can partly be absorbed and partly be reflected
by the body and this absorption and reflection of radiation depends on the color of
body.
• Thermal radiation travels through vacuum on straight line and with the velocity of
light.
• Thermal radiations can be reflected and refracted.
Black Body Radiation
• A body that absorbs all the radiation falling on it is called a black body.
• Radiation emitted by black body is called Black Body radiation.
• A black body is also called an ideal radiator.
• For practical purpose black body can be considered as an enclosure painted black
from inside and a small hole is made in the wall.
• Once radiation enters the enclosure it has very little chance to come out of the hole
and it gets absorbed after multiple refractions inside the closure.
• Concept of a perfect black body is an ideal one.
Stefan Boltzmann law
46
• The rate Urad at which an object emits energy via EM radiation depends on objects
surface area A and temperature T in kelvin of that area and is given by 4
radU = ATσε (2)
where
8 425.6702 10 Wx K
mσ −=
is stefan boltzmann constant and ε is emissivity of object's surface with value
between 0 and 1.
• Black - Body radiator has emissivity of 1.0 which is an ideal limit and does not
occur in nature.
• The rate Uabs at which an object absorbs energy via thermal radiation from its
environment iwth temperature
Tenv (in kelvin) is
Uzbs = σεA(Tenv)4 (3)
where ε is same as in equation 2
• Since an object radiate energy to the environment and absorb energy from
environment its net energy exchange due to thermal radiation is
U=Uabs . Urad
= σεA{(Tenv)4.T4} (4)
• u is positive if net energy is being absorbed via radiation and negative if it is being
lost via radiation.
Nature of thermal Radiation
• Radiation emitted by a black body is a mixture of waves of different wavelengths
and only a small range of wavelength has significant contribution in the total
radiation.
• A body is heated at different temperature and Energy of radiation is plotted against
wavelength is plotted for different temperature we get following curves.
• These curves show
(i) Energy is not uniformly distributed in the radiation spectrum of black body.
47
(ii) At a given temperature the intensity of radiations increases with increase in
wavelength, becomes maximum at particular wavelength and further increase
in wavelength leads to decrease in intensity of heat radiation.
(iii) Increase in temperature causes increase in energy emission for all
wavelengths.
(iv) Increase in temperature causes decrease in λm, where λm is wavelength
corresponding to highest intensity. This wavelength λm is inversely
preoperational to the absolute temperature of the emitter.
λm T = b (5)
where b is a constant and this equation is known as Wein's displacement law.
b = 0.2896 x 10-2mk for black body and is known as Wien's constant.
Kirchoff's law
• Good absorbers of radiation are also good radiators this statement is quantitatively
explained by Kirchoff's law.
(i) Emissive Power - Emissive power denotes the energy radiated per unit area
per unit solid angle normal to the area.
( ) ( )( )E u / A tω= ∆ ∆ ∆ ∆
where, ∆u is the energy radiated by area ∆A of surface in solid angle ∆ω in
time ∆t.
(ii) Absorptive Power - Absorptive power of a body is defined as the fraction of
the incident radiation that is absorbed by the body a(absorptive power) =
energy absorbed /energy incident.
(iii) Kirchoff's Law - It states that at any given temperature the ratio of emissive
power to the absorptive power is constant for a bodies and this constant is
equal to the emissive power of perfect B.B. at the same temperature.
E/abody = EB.B
• From kirchoff's law we can say that a body having high emissive power should
have high absorptive power and those having low emissive power should have law
absorptive power so as to keep the ratio E/a same.
Newton's Law of Cooling
• Considered a hot body at temperature T1 is placed in surrounding at temperature T2.
• For small temperature difference between the body and surrounding rate of cooling
is directly proportional to the temperature difference and surface area exposed i.e.,
dT/dt = -bA(T1-T2)
• This is known a Newton's law of cooling.
b depends on nature of surface involved and the surrounding conditions. Negative
sign is to indicate that T1>T2, dT/dt is negative and temperature decreases with
time.
• According to this law, the rate of cooling is directly proportional to the excess of
temperature
48
Supplementary Material on Additional Sub-Topics
included in Class XII Physics Theory Syllabus in the
Chapter on Communication Systems.
A. INTERNET
Introduction
Invention of computers changed the working style of people in twentieth century. Its
capability to tirelessly and sequentially do arithmetical and logical operations made
the human life simpler and faster. Offices, universities, banks, schools etc. nothing
remained unaffected by use of computers. This was not enough and before the end of
twentieth century we succeeded in creating a global network of computers that
provides ways to exchange information and to communicate among all computers
connected to the network. This global network of computers is what we now call
Internet (or simply net). Internet, in fact, is the short form of INTER-NET work which
is the interconnected network of all worldwide servers. Networking of computers: The
way Internet works Two or more than two computers are said to be networked when
they are able to exchange information between them. This sharing of information can
be through wires connecting these computers or some wireless means of
communications like Wi-Fi. Networking of computers at small scale (e.g. within an
office, a building or a school) is called Local Area networking (LAN). One can also
connect devices like printer, scanner, etc. to a LAN as shown in the figure below.
Networking of computers: The way Internet works
Two or more than two computers are said to be networked when they are able to
exchange information between them. This sharing of information can be through
wires connecting these computers or some wireless means of communications like
Wi-Fi.
A local area network (LAN) is a computer
network that interconnects computers
within a limited area such as a home,
school, computer laboratory, or office
building, using network media.[1] The
defining characteristics of LANs, in contrast
to wide area networks (WANs), include their
smaller geographic area, and non-inclusion
of leased telecommunication lines.[citation needed]
ARCNET, Token Ring and other technology
A Local Area Network (LAN)
49
standards have been used in the past, but Ethernet over twisted pair cabling,
and Wi-Fiare the two most common technologies currently used to build LANs.
Most LANs connect workstations and personal computers. Eachnode (individual
computer ) in a LAN has its own CPU with which itexecutes programs, but it also is
able to access data and devicesanywhere on the LAN. This means that
many users can share expensive devices, such as laser printers, as well as data.
Users can also use the LAN to communicate with each other, by sending e-mail or
engaging in chat sessions.
LANs are capable of transmitting data at very fast rates, much faster than data can
be transmitted over a telephone line; but the distances are limited, and there is also a
limit on the number of computers that can be attached to a single LAN.
Types of Local-Area Networks (LANs)
There are many different types of LANs, with Ethernets being the most common
for PCs. Most Apple Macintoshnetworks are based on Apple's AppleTalk network
system, which is built into Macintosh computers.
The following characteristics differentiate one LAN from another:
Topology : The geometric arrangement of devices on the network. For example,
devices can be arranged in a ring or in a straight line.
Protocols : The rules and encoding specifications for sending data. The protocols
also determine whether the network uses a peer-to-peer or client/server architecture.
Media : Devices can be connected by twisted-pair wire, coaxial cables, or fiber
optic cables. Some networks do without connecting media altogether, communicating
instead via radio waves.
A wide area network (WAN) is a network that
covers a broad area (i.e.,
any telecommunications network that links
across metropolitan, regional, national or
international boundaries) using leased
telecommunication lines. Business and
government entities use WANs to relay data
among employees, clients, buyers, and
suppliers from various geographical locations.
In essence, this mode of telecommunication
allows a business to effectively carry out its
daily function regardless of location. The
Internet can be considered a WAN as well, and
is used by businesses, governments, organizations, and individuals for almost any
purpose imaginable.[1]
Formation of a WAN
50
Related terms for other types of networks are personal area networks (PANs), local
area networks (LANs), campus area networks (CANs), or metropolitan area
networks (MANs) which are usually limited to a room, building, campus or specific
metropolitan area (e.g., a city) respectively.
WANs are used to connect LANs and other types of networks together, so that users
and computers in one location can communicate with users and computers in other
locations. Many WANs are built for one particular organization and are private.
Others, built by Internet service providers, provide connections from an
organization's LAN to the Internet. WANs are often built using leased lines. At each
end of the leased line, a router connects the LAN on one side with a
second router within the LAN on the other. Leased lines can be very expensive.
Instead of using leased lines, WANs can also be built using less costly circuit
switching or packet switching methods.
One can built such a local area network of computers within an institution by
connecting all or some of their computers. Every LAN has some main computers
called server computers. These servers are used to connect LAN to other networks
through telephone lines or satellites. In this way, by connecting various LAN, a Wide
Area Network (WAN) is created as shown in the figure.
Various interlinked WAN together constitute what we call an Internet as shown in the
figure.
A metropolitan area network (MAN) is a computer network larger than a local area
network, covering an area of a few city blocks to the area of an entire city, possibly
also including the surrounding areas.[1]
A MAN is optimized for a larger geographical area than a LAN, ranging from several
blocks of buildings to entire cities. MANs can also depend on communications
channels of moderate-to-high data rates. A MAN might be owned and operated by a
single organization, but it usually will be used by many individuals and
organizations. MANs might also be owned and operated as public utilities. They will
often provide means for inter networking of local networks.
“A Metropolitan Area Network (MAN) is a large computer network that spans a
metropolitan area or campus. Its geographic scope falls between a WAN and LAN.
MANs provide Internet connectivity for LANs in a metropolitan region, and connect
them to wider area networks like the Internet.”
All information related to a local network is stored in server computers of LAN. The
servers of every LAN act as channel for information exchange between computers
connected to LAN and also servers of other networks. Every computer that extracts
information from a server is called a client computer.
51
Formation of Internet
First network of computers called Advanced Research Projects Agency NETwork
(ARPANET) was developed by US Department of Defense in 1969. By 1990, many
countries of the world came up with a common set of rules for Internet communication
among computers called protocols.
Nowadays, standard sets of protocols called Transmission Control Protocol/Internet
Protocol (TCP/IP) are used for exchange of information through Internet.
It is important to note that the exchange of information on Internet is very fast (at the
speed of light) as electronic signals (messages) of computers are communicated
through electromagnetic waves.
In India, Internet was started in November 1988 by VSNL (Videsh Sanchar Nigam
Limited) in Mumbai.
On Internet, information is provided / available through webpages that may contain
text, images, videos, etc. One can move from one webpage on Internet to another
through a system called interlinked hypertext documents. In this system, one webpage
is linked to another webpage by providing hyperlinking (a way of highlighting) to any
text, image or video. This way of accessing information on Internet through
interlinking of webpage is called www or World Wide Web.
Anyone can provide specific information on Internet by making a couple of webpages
containing that information. Such a set of webpages together constitute a website. One
can design a website of own organization containing information about its different
aspects and its activities.
Anyone can connect its computer to the Internet network through various Internet
Service Providers (ISP) by paying a prescribed fee. Commonly, mobile network
companies also acts as ISP.
Applications of Internet
People use Internet for many purposes like searching and viewing information on any
topic of interest, for sending electronic mails (e-mails), for e-banking, e-shopping
52
(e-commerce), e-booking (e-ticketing) etc. This list of uses of Internet is endless.
(i) Internet Surfing: Moving on Internet from one webpage/website to another is called
Internet surfing. It is an interesting way of searching and viewing information on any
topic of interest.
(ii) E-mail: E-mail means “electronic mail”. This is the most used application of
Internet. E-mail is a way of sending texts written on computers through Internet.
Along with text one can send images and videos too. This is cheapest and fastest way
of sending messages.
For using this facility of Internet one needs to create a personal email account with an
email-Id (identity) or email address. Email-Id is like an identity card (Name and
address) through which people can identify and communicate to you through Internet.
A few websites provides free email accounts and Ids to Internet users. These email-Ids
are password protected and thus no one other than whom it belongs can use them.
Internet Service Providers (ISP) also provide email-Ids.
Every email-Id has two parts separated by a sign @ called at the rate of. For example,
“[email protected]” is an email Id. Its two parts:
(i) Part before @ sign : prakrittyagi
It is personal information part. Here it denotes a name Prakrit Tyagi.
(ii) Part after @ sign : gmail.com
It is called a domain name. It provides information about the server that is providing
this email facility.
A message sent through email is instantly delivered to the addressee because
communication of messages is by means of electromagnetic waves through Internet.
Beauty of email lies in the fact that a message is stored in an email account even at a
time when its user is not connected to Internet, which can be viewed later.
Email’s use is increasing day by day. We can even send personalized greeting cards
through email. Today it has become an extremely popular communication tool.
(iii) E-banking: It is an electronic payment system allowing customers to proceed for
financial transactions on a website operated by that financial institution (usually a
bank). For this purpose, customer needs to be a member of that institution, needs to
have internet access and must register with the institution for the service. In turn, the
financial institution provides the login number and password to the customer for
his/her unique identification. With the help of this facility a customer can link his
account with any other facilities such as checking on line status of the balance, check
book requisition, loan, recurring, credit card, debit card etc.
(iv) E-shopping (E-commerce): Virtual Malls are available on Internet where one can
view and order to purchase various products. Buying products through product selling
53
websites is called e-shopping. These websites provide the buyer pay cash on delivery
or making online payments (using e-banking or credit cards) options. Similarly, there
are websites on which one can upload (put) photographs of products, which you want
to sell. This trading of products using Internet along with many other market related
activities is called e-commerce.
(v) E-booking (e-ticketing): It is an application developed for ticket reservation
through the Internet to help the travel and tourism industry. It helps consumers to
book flight tickets, railway tickets, hotels, holiday packages, insurance and other
services online.
To book an e-ticket, a customer needs to visit a home page of an Airline Company or
Indian Railways. Once he/she enters the travel preference, gets an opportunity to view
the available flights / trains through an appropriate interface. Once the choice is fixed,
the customer needs to select the mode for transfer of money. Once the payment is
done through an authentic mode (like e-banking), an online ticket is issued to the
customer.
(vi) Social Networking: It is a service providing a platform to the people having same
interests to build a social network. It is a web-based service allows an individual to
create his/her own profile, list of users with whom they want to connect. The service
allows the user to share their ideas, pictures, events, activities etc. with their group.
Facebook, Twitter, Google+ etc. are some popular social networking sites.
B. MOBILE TELEPHONY
Introduction
As we look around, in markets, on trains and buses, people crossing streets, we can
see many individuals talking on cell phones or mobile phones. Mobile phones have
changed the way we live and communicate. With
advancement of technology, look and utility of mobile phone
has also undergone change. In latest mobile phones, along
with making and receiving phone calls one can also:
• Store contact information
• Make task or to-do lists
• Keep track of appointments and set reminders
• Use the built-in calculator for simple math
• Send or receive e-mail
• Get information (news, entertainment, stock quotes) from
the Internet
• Play games
54
• Listen radio/music and watch TV
• Send text messages
• Take photos and videos etc.
In a way, today’s mobile phone is a handy computer equipped with Internet.
In ordinary landline phones, phone instruments are connected to a telephone
exchange through electric wires, which in turn connect our phone calls to the other
phones. However, wire connections limit the mobility of a landline phone. Mobile
phone technology has successfully overcome this limitation.
Mobile phone is a low power operated device (transmitter), which can wirelessly send
and receive radio frequency signals. Before this, walkie-talkie was also a wireless
system of communication. You must have seen a policeman talking on his wireless
set. After completing one sentence, he says “Over” and then listens. This was because
the same radio frequency is used for both sending and receiving the audio signal.
However, in a mobile phone, the outgoing and incoming signals use different
frequencies, so the two individuals can talk and listen at the same time.
Working principle of Mobile phone
In a mobile phone, it is possible to talk while
moving. This becomes possible because of a
cellular radio network technology (a
replacement of telephone exchange system).
Under a cellular radio network a given physical
area is divided into smaller parts call cells (or
cell zones). To completely cover a given area use
of hexagonal cells is a best possible way as
shown in the figure.
In every hexagonal cell a radio antenna is
installed to receive and send radio signals to
and from mobile phones physically present
within the cell. All cell antennas present within
an area are connected to each other through a
network (the way computers are connected in internet).
All network related works including handling of all the incoming and outgoing calls are
managed by a central control room called Mobile Telephone Switching Office (MTSO).
i.e. MTSO is basically a telephone exchange for mobile phone calls.
Every cell antenna has a working range of minimum 1.5 to 2 km and maximum up to
48 to 56 or more km area around it. When a mobile phone is switched on, MTSO
records its location by identifying the cell in which it is present. When a mobile phone
55
user moves from one cell zone to another cell zone, MTSO of its own switches mobile
phone link to new cell antenna. This way, user gets an uninterrupted link to talk while
on move. Also, mobile phones use high frequency radio waves for conversation. Audio
signals of these waves are better. As mobile phones works on cell division of physical
areas they are also referred as cell phones.
Scientific process of a mobile phone call
When we dial a mobile number from your mobile phone, an oscillator circuit
(frequency generator) inside the mobile generates a particular frequency
electromagnetic wave. This electromagnetic wave carrying called number’s information
is transmitted through antenna of your mobile to the antenna of the cell in which we
are present. The cell antenna in turn transfers this signal to MTSO. The MTSO
computer system identifies the location (cell) of the mobile phone you have dialed and
connects you to that phone. The caller mobile on receiving your signal generates again
through an oscillator circuit your ID (mobile number) and displays it. This whole
process happens with in a few seconds as all the signals are transferred through
electromagnetic waves, which travel at the speed of light. Here, it is important to note
that mobile phone call is transferred from dialer cell antenna to MTSO and MTSO to
caller cell antenna only through cell antenna lying in between. That is why mobile
phone network is also called terrestrial cellular network.
56
Mobile phone numbering system
Due to mobility of a mobile phone it is necessary to
identify every mobile phone. For this, a SIM (Subscriber
Identity Module) card in inserted in every mobile phone.
SIM card is like an identity card of its user. It is a small
IC (Integrated circuit) chip with a unique SIM number and
a mobile phone number. A typical SIM card is shown in
the figure. All SIM cards are issued by mobile operator
companies and their information is provided to MTSO.
After SIM verification, MTSO activates the mobile number
of the user. This makes a mobile phone usable. Every
mobile number in India is of 10 digits. All mobile numbers in India have the prefix 9,8
or 7. As per National Numbering Plan 2003, the way to split mobile numbers is as
XXXX-NNNNNN where XXXX is Network operator digits and NNNNNN is the subscriber
number digits. To regulate the use of mobile phones system in India a Telecom
Regulatory Authority of India (TRAI) was established in 1997 by an act of Parliament.
Mobile network Generations (1G, 2G, 3G & 4G)
With increasing use of mobile phones and advancement of technology, it is pertinent
to make the mobile phone networks more efficient. The efficiency of mobile networks is
mentioned by word ‘Generation’ and abbreviation ‘G’. 1G were first generation of
mobile networks, which were based on analogue radio signals. 2G were narrow band
digital signal based networks with good quality of calls. They provided world over
connectivity. 3G networks increased the data transfer speed for efficient use of
Internet on mobile phone. 4G networks are going to provide a high-speed internet
facility on mobile phones for surfing net, chatting, viewing television, listening music
etc.
C. Global Positioning System
Since ages man has invented various instruments to assist him in navigation on earth.
Magnetic compass is one of the oldest navigational instrument man has been using for
many centuries for direction identification on earth’s surface. Global Positioning
System (GPS) based devices are the latest navigation assistance devices used these
days. GPS devices provide accurate real time location and much more information to
its user for easier and comfortable navigation even through his or her local streets. A
commonly available GPS device is shown in the figure. When fitted in a car, it shows
speed of the car, time, longitude coordinates and map of near-by area.
57
What is Global Positioning System (GPS).
Global positioning system GPS is a method
of identifying location or position of any
point (or a person) on earth using a system
of 24 satellites, which are continuously
orbiting, observing, monitoring and mapping
the earth surface. Every such satellite
revolves around earth twice a day at a
distance of about 20000 km from it. The
given figure shows sketch view of 24 GPS
satellites orbiting around the earth.
The orbits of these satellites are so aligned
that at least four of them always keep looking any given point on earth surface. This is
minimum necessary requirement for correct and accurate location identification
through this system. In the given figure, the given location at the instant is visible to
12 satellites.
Working principle of a GPS device
For using the GPS system of satellites, a person needs a GPS device fitted with a
transmitter/receiver for sending/receiving signals (electromagnetic radio waves) so
that it can link up with GPS satellites in real time.
The unique location (or longitude coordinates) of a GPS user is determined by
measuring its distance from at least three GPS satellites. Based on these distance
measurements, the location calculations are done by the microprocessor (computing
device) fitted in the GPS device.
58
For measuring distance of three GPS satellites from the GPS device user, the time
taken by a radio signal to travel from device to satellites and back are recorded by the
GPS device.
For, example, if a radio signal takes 0.140 seconds to travel back from a satellite-1 to
its GPS user.
Then, Distance of satellite-1 from user = (Speed of light x Time)
2
83 x 10 m/s x 0.140s
2
42000
2
km
=
=
= 21000 km
Following the above method, let D1, D2 and D3 be the distances of three satellites
from a GPS device user. From this information, the identification of unique location of
the GPS device user is done as follows:
(1) If user is at a distance “D1” from satellite-1. Then user’s location can be anywhere
on a circumference of circle of radius “D1” from satellite-1 as shown in fig below.
(2) If user is at a distance “D2” from satellite-2. Then user’s
location can be either at intersecting points X or Y of
circumferences of circles of radius D1 and D2 from satellite-1
and 2 respectively as shown in figure below.
59
(3) If user is at a distance “D3” from satellite-3.
Then user’s location will be at the intersecting point
of circumferences of circles of radius D1, D2 and
D3 from satellite-1, 2 and 3 respectively as shown
in fig(c). i.e. here user is at point Y.
This way, minimum three satellites together provide the exact location (longitude
coordinates) of the GPS device user on his display board.
If a person is at some height on earth surface, then using distance information from
minimum 4-GPS satellites even altitude of the user can also be measured.
It may be noted that since all 24-GPS satellites orbit in predefined orbits, therefore
their locations are precisely predetermined. It is these known locations of 3 or 4 -GPS
satellites (3 or 4 sets of longitude coordinates) and their distances to GPS device that
assist a GPS user (i.e. its computing device) in locating its own longitude coordinates.
Applications of GPS
• Global positioning system has many day-to-day applications:
• It helps in navigation on water, air and land.
• It assists in map designing of a location.
• It helps automatic vehicle movements (without man)
• One can measure speed of moving object using this technology.
• One can locate change in position of glaciers, mountains heights.
• It assists in keeping standard time world over.
• It assists in tracking animals and birds and studying their movements by attaching
GPS devices to their bodies.
• It assists in airplane traffic movement.
• It assists visually impaired in location identification.
Now-a-days, various devices like mobile phones, i-pad etc. come equipped with
pre-loaded geographical maps and GPS software which identifies the location of these
devices using GPS system. GPS is a free service available to anyone in the world with a
GPS device.
60
Units, Dimensions and Measurement / 61
1. Units, Dimensions and Measurement
1.1 Physical Quantity
A quantity which can be measured and expressed in form of laws is called a physical quantity. Physical quantity (Q) = Magnitude × Unit = n × u Where, n represents the numerical value and u represents the unit. as the unit(u) changes,
the magnitude(n) will also change but product ‘nu’ will remain same. i.e. n u = constant, or n1u1 = n2u2 = constant;
1.2 Fundamental and Derived Units
Any unit of mass, length and time in mechanics is called a fundamental, absolute or base unit. Other units which can be expressed in terms of fundamental units, are called derived units
System of units: A complete set of units, both fundamental and derived for all kinds of physical quantities is called system of units.
(1) CGS system (2) MKS system (3) FPS system. (4) S.I. system: It is known as International system of units. There are seven fundamental
quantities in this system. These quantities and their units are given in the following table.
Quantity Name of Unit Symbol
Length metre m
Mass kilogram kg
Time second s
Electric Current ampere A
Temperature Kelvin K
Amount of Substance mole mol
Luminous Intensity candela cd
Besides the above seven fundamental units two supplementary units are also defined – Radian (rad) for plane angle and Steradian (sr) for solid angle.
1.3 Practical Units
(1) Length: (i) 1 fermi = 1 fm = 10–15 m (ii) 1 X-ray unit = 1XU = 10–13 m (iii) 1 angstrom = 1Å = 10–10 m = 10–8 cm = 10–7 mm = 0.1 µmm (iv) 1 micron = µm = 10–6 m (v) 1 astronomical unit = 1 A.U. = 1. 49 × 1011 m ≈ 1.5 × 1011 m ≈ 108 km (vi) 1 Light year = 1 ly = 9.46 × 1015 m (vii) 1 Parsec = 1pc = 3.26 light year
(2) Mass: (i) Chandra Shekhar unit: 1 CSU = 1.4 times the mass of sun = 2.8 × 1030 kg (ii) Metric tonne: 1 Metric tonne = 1000 kg (iii) Quintal: 1 Quintal = 100 kg
62 /Academic Excellence in Physics
(iv) Atomic mass unit (amu): amu = 1.67 × 10–27 kg mass of proton or neutron is of the order of 1 amu
(3) Time: (i) Year: It is the time taken by earth to complete 1 revolution around the sun in its orbit. (ii) Lunar month: It is the time taken by moon to complete 1 revolution around the
earth in its orbit. 1 L.M. = 27.3 days (iii) Solar day: It is the time taken by earth to complete one rotation about its axis with
respect to sun. Since this time varies from day to day, average solar day is calculated by taking average of the duration of all the days in a year and this is called Average Solar day.
1 Solar year = 365.25 average solar day
or average solar day =1
365.25 the part of solar year
(iv) Sedrial day: It is the time taken by earth to complete one rotation about its axis with respect to a distant star.
1 Solar year = 366.25 Sedrial day = 365.25 average solar day Thus 1 Sedrial day is less than 1 solar day.
(v) Shake: It is an obsolete and practical unit of time. 1 Shake = 10– 8 sec
1.4 Dimensions of a Physical Quantity
When a derived quantity is expressed in terms of fundamental quantities, it is written as a product of different powers of the fundamental quantities. The powers to which fundamental quantities must be raised in order to express the given physical quantity are called its dimensions.
1.5 Important Dimensions of Complete Physics
Mechanics
S. N. Quantity Unit Dimension
(1) Velocity or speed (v) m/s [M0L1T –1] (2) Acceleration (a) m/s2 [M0LT –2] (3) Momentum (P) kg-m/s [M1L1T –1] (4) Impulse (I) Newton-sec or kg-m/s [M1L1T –1] (5) Force (F) Newton [M1L1T –2] (6) Pressure (P) Pascal [M1L–1T –2] (7) Kinetic energy (EK) Joule [M1L2T –2] (8) Power (P) Watt or Joule/s [M1L2T –3] (9) Density (d) kg/m3 [M1L– 3T 0]
(10) Angular displacement (θ) Radian (rad.) [M0L0T 0]
(11) Angular velocity (ω) Radian/sec [M0L0T – 1]
(12) Angular acceleration (α) Radian/sec2 [M0L0T – 2]
(13) Moment of inertia (I) kg-m2 [M1L2T0]
(14) Torque (τ) Newton-meter [M1L2T –2]
Units, Dimensions and Measurement / 63
S. N. Quantity Unit Dimension
(15) Angular momentum (L) Joule-sec [M1L2T –1] (16) Force constant or spring constant (k) Newton/m [M1L0T –2] (17) Gravitational constant (G) N-m2/kg2 [M–1L3T – 2] (18) Intensity of gravitational field (Eg) N/kg [M0L1T – 2] (19) Gravitational potential (Vg) Joule/kg [M0L2T – 2] (20) Surface tension (T) N/m or Joule/m2 [M1L0T – 2] (21) Velocity gradient (Vg) Second–1 [M0L0T – 1]
(22) Coefficient of viscosity (η) kg/m-s [M1L– 1T – 1]
(23) Stress N/m2 [M1L– 1T – 2] (24) Strain No unit [M0L0T 0] (25) Modulus of elasticity (E) N/m2 [M1L– 1T – 2]
(26) Poisson Ratio (σ) No unit [M0L0T 0]
(27) Time period (T) Second [M0L0T1] (28) Frequency (n) Hz [M0L0T –1]
Heat
S. N. Quantity Unit Dimension
(1) Temperature (T) Kelvin [M0L0T0θ 1] (2) Heat (Q) Joule [ML2T– 2]
(3) Specific Heat (c) Joule/kg-K [M0L2T– 2θ –1] (4) Thermal capacity Joule/K [M1L2T – 2θ –1] (5) Latent heat (L) Joule/kg [M0L2T – 2]
(6) Gas constant (R) Joule/mol-K [M1L2T– 2θ – 1]
(7) Boltzmann constant (k) Joule/K [M1L2T– 2θ – 1]
(8) Coefficient of thermal conductivity (K)
Joule/m-s-K [M1L1T– 3θ – 1]
(9) Stefan's constant (σ) Watt/m2-K4 [M1L0T– 3θ – 4] (10) Wien's constant (b) Meter-K [M0L1Toθ1] (11) Planck's constant (h) Joule-s [M1L2T–1]
(12) Coefficient of Linear Expansion ()
Kelvin–1 [M0L0T0θ –1]
(13) Mechanical eq. of Heat (J) Joule/Calorie [M0L0T0] (14) Vander wall’s constant (a) Newton-m4 [ML5T– 2] (15) Vander wall’s constant (b) m3 [M0L3T0]
Electricity
S. N. Quantity Unit Dimension
(1) Electric charge (q) Coulomb [M0L0T1A1] (2) Electric current (I) Ampere [M0L0T0A1] (3) Capacitance (C) Coulomb/volt or Farad [M–1L– 2T4A2] (4) Electric potential (V) Joule/coulomb M1L2T–3A–1
64 /Academic Excellence in Physics
S. N. Quantity Unit Dimension
(5) Permittivity of free space (ε0) 2
2–
Coulomb
Newton meter [M–1L–3T4A2]
(6) Dielectric constant (K) Unitless [M0L0T0] (7) Resistance (R) Volt/Ampere or ohm [M1L2T– 3A– 2]
(8) Resistivity or Specific resistance (ρ)
Ohm-meter [M1L3T– 3A– 2]
(9) Coefficient of Self-induction (L)
−volt second
ampereor henery or ohm-second [M1L2T– 2A– 2]
(10) Magnetic flux (φ) Volt-second or weber [M1L2T–2A–1]
(11) Magnetic induction (B) −
newton
ampere meter − 2
Joule
ampere meter
−2
secondvolt
meter or Tesla
[M1L0T– 2A– 1]
(12) Magnetic Intensity (H) Ampere/meter [M0L– 1T0A1] (13) Magnetic Dipole Moment (M) Ampere-meter2 [M0L2T0A1]
(14) Permeability of Free Space (µ0)
2
Newton
ampere or
−2
Joule
ampere meteror
−
−
secondVolt
ampere meteror
− secOhm ond
meter or
henery
meter
[M1L1T–2A–2]
(15) Surface charge density (σ) 2−metreCoulomb [M0L–2T1A1]
(16) Electric dipole moment (p) meterCoulomb − [M0L1T1A1]
(17) Conductance (G) (1/R) 1−ohm [M–1L–2T3A2]
(18) Conductivity (σ) (1/ρ) 11 −− meterohm [M–1L–3T3A2]
(19) Current density (J) Ampere/m2 M0L–2T0A1
(20) Intensity of electric field (E) Volt/meter, Newton/coulomb M1L1T –3A–1
(21) Rydberg constant (R) m–1 M0L–1T0
1.6 Quantities Having Same Dimensions.
S. N. Dimension Quantity
(1) [M0L0T–1] Frequency, angular frequency, angular velocity, velocity gradient and decay constant
(2) [M1L2T–2] Work, internal energy, potential energy, kinetic energy, torque, moment of force
(3) [M1L–1T–2] Pressure, stress, Young’s modulus, bulk modulus, modulus of rigidity, energy density
(4) [M1L1T–1] Momentum, impulse
(5) [M0L1T–2] Acceleration due to gravity, gravitational field intensity
Units, Dimensions and Measurement / 65
(6) [M1L1T–2] Thrust, force, weight, energy gradient
(7) [M1L2T–1] Angular momentum and Planck’s constant
(8) [M1L0T–2] Surface tension, Surface energy (energy per unit area)
(9) [M0L0T0] Strain, refractive index, relative density, angle, solid angle, distance gradient, relative permittivity (dielectric constant), relative permeability etc.
(10) [M0L2T–2] Latent heat and gravitational potential
(11) [M0L2T–2θ–1] Thermal capacity, gas constant, Boltzmann constant and entropy
(12) [M0L0T1] , ,l g m k R g , where l = length
g = acceleration due to gravity, m = mass, k = spring constant
(13) [M0L0T1] L/R, LC , RC where L = inductance, R = resistance, C = capacitance
(14) [ML2T–2]
2 22 2 2, , , , , ,
V qI Rt t VIt qV LI CV
R C where I = current, t = time, q = charge,
L = inductance, C = capacitance, R = resistance
1.7 Application of Dimensional Analysis.
(1) To find the unit of a physical quantity in a given system of units. (2) To find dimensions of physical constant or coefficients. (3) To convert a physical quantity from one system to the other. (4) To check the dimensional correctness of a given physical relation: This is based on the
‘principle of homogeneity’. According to this principle the dimensions of each term on both sides of an equation must be the same.
(5) To to derive new relations.
1.8 Limitations of Dimensional Analysis.
(1) If dimensions are given, physical quantity may not be unique. (2) Numerical constant having no dimensions cannot be deduced by the methods of
dimensions. (3) The method of dimensions can not be used to derive relations other than product of
power functions. For example, 2(1 / 2)s ut at= + or siny a t= ω
(4) The method of dimensions cannot be applied to derive formula consist of more than 3 physical quantities.
1.9 Significant Figures.
Significant figures in the measured value of a physical quantity tell the number of digits in which we have confidence. Larger the number of significant figures obtained in a measurement, greater is the accuracy of the measurement. The reverse is also true.
The following rules are observed in counting the number of significant figures in a given measured quantity.
(1) All non-zero digits are significant. (2) A zero becomes significant figure if it appears between to non-zero digits. (3) Leading zeros or the zeros placed to the left of the number are never significant. Example: 0.543 has three significant figures. 0.006 has one significant figures.
66 /Academic Excellence in Physics
(4) Trailing zeros or the zeros placed to the right of the number are significant. Example: 4.330 has four significant figures. 343.000 has six significant figures. (5) In exponential notation, the numerical portion gives the number of significant figures. Example: 1.32 × 10–2 has three significant figures.
1.10 Rounding Off
(1) If the digit to be dropped is less than 5, then the preceding digit is left unchanged. Example: 82.7=x is rounded off to 7.8, again 94.3=x is rounded off to 3.9. (2) If the digit to be dropped is more than 5, then the preceding digit is raised by one. Example: x = 6.87 is rounded off to 6.9, again x = 12.78 is rounded off to 12.8. (3) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit
is raised by one. Example: x = 16.351 is rounded off to 16.4, again x = 6.758 is rounded off to 6.8. (4) If digit to be dropped is 5 or 5 followed by zeros, then preceding digit is left unchanged, if
it is even. Example: x = 3.250 becomes 3.2 on rounding off, again x = 12.650 becomes 12.6 on
rounding off. (5) If digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one,
if it is odd. Example: x = 3.750 is rounded off to 3.8, again x = 16.150 is rounded off to 16.2.
1.11 Significant Figures in Calculation.
The following two rules should be followed to obtain the proper number of significant figures in any calculation.
(1) The result of an addition or subtraction in the number having different precisions should be reported to the same number of decimal places as are present in the number having the least number of decimal places
(2) The answer to a multiplication or division is rounded off to the same number of significant figures as is possessed by the least precise term used in the calculation.
1.12 Order of Magnitude.
Order of magnitude of quantity is the power of 10 required to represent the quantity. For determining this power, the value of the quantity has to be rounded off. While rounding off, we ignore the last digit which is less than 5. If the last digit is 5 or more than five, the preceding digit is increased by one. For example,
(1) Speed of light in vacuum = 3 × 108 ms–1 ≈ 108 m/s (ignoring 3 < 5) (ignoring 3 < 5) (2) Mass of electron = 9.1 × 10–31 kg 10–30 kg (as 9.1 > 5).
1.13 Errors of Measurement.
The measured value of a quantity is always somewhat different from its actual value, or true value. This difference in the true value of a quantity is called error of measurement.
(1) Absolute error: Absolute error in the measurement of a physical quantity is the magnitude of the difference between the true value and the measured value of the quantity.
Let a physical quantity be measured n times. Let the measured value be a1, a2, a3, ….. an.
The arithmetic mean of these value is + +
= 1 2 .... nm
a a aa
n
Units, Dimensions and Measurement / 67
Usually, am is taken as the true value of the quantity, if the same is unknown otherwise. By definition, absolute errors in the measured values of the quantity are ∆a1 = am – a1 ∆a2 = am – a2 …………. ∆an = am – an The absolute errors may be positive in certain cases and negative in certain other cases. (2) Mean absolute error: It is the arithmetic mean of the magnitudes of absolute errors in
all the measurements of the quantity. It is represented by ∆ .a Thus
∆ + ∆ + ∆
∆ = 1 2| | | | .....| |na a aa
n
Hence the final result of measurement may be written as = ± ∆ma a a
This implies that any measurement of the quantity is likely to lie between + ∆( )ma a and
− ∆( ).ma a (3) Relative error or Fractional error: Relative error or Fractional error
∆
= =mean absolute error
mean value m
a
a
(4) Percentage error: Percentage error ∆
= ×100%m
a
a
1.14 Propagation of Errors
(1) Error in sum of the quantities: Suppose x = a + b Let ∆a = absolute error in measurement of a ∆b = absolute error in measurement of b ∆x = absolute error in calculation of x i.e. sum of a and b. The maximum absolute error in x is ∆ = ± ∆ + ∆( )x a b (2) Error in difference of the quantities: Suppose x = a – b The maximum absolute error in x is ∆ = ± ∆ + ∆( )x a b (3) Error in product of quantities: Suppose x = a × b
The maximum fractional error in x is ∆ ∆ ∆
= ± +
x a b
x a b
(4) Error in division of quantities: Suppose =a
xb
The maximum fractional error in x is ∆ ∆ ∆
= ± +
x a b
x a b
(5) Error in quantity raised to some power: Suppose =n
m
ax
b
The maximum fractional error in x is ∆ ∆ ∆
= ± +
x a bn m
x a b
� The quantity which have maximum power must be measured carefully because it's contribution to error is maximum.
68 / Academic Excellence in Physics
2. Kinematics
2.1 Motion in One Dimension: Position
Position of any point is completely expressed by two factors: Its distance from the observer and its direction with respect to observer.
That is why position is characterised by a vector known as position vector. Let point P is in a xy plane and its coordinates are (x, y). Then position vector ( )r
� of point will
be ˆ ˆxi yj+ and if the point P is in a space and its coordinates are (x, y, z) then position vector can
be expressed as ˆˆ ˆ .r xi yj zk= + +�
2.2 Rest and Motion.
If a body does not change its position as time passes with respect to frame of reference, it is said to be at rest.
And if a body changes its position as time passes with respect to frame of reference, it is said to be in
motion.
Frame of Reference: It is a system to which a set of coordinates are attached and with reference to which observer describes any event.
Rest and motion are relative terms. It depends upon the frame of references.
2.3 Types of Motion.
One dimensional Two dimensional Three dimensional
Motion of a body in a straight line is called one dimensional motion.
Motion of body in a plane is called two dimensional motion.
Motion of body in a space is called three dimensional motion.
When only one coordinate of the position of a body changes with time then it is said to be moving one dimensionally.
When two coordinates of the position of a body changes with time then it is said to be moving two dimensionally.
When all three coordinates of the position of a body changes with time then it is said to be moving three dimensionally.
e.g.. Motion of car on a straight road. Motion of freely falling body.
e.g. Motion of car on a circular turn. Motion of billiards ball.
e.g.. Motion of flying kite. Motion of flying insect.
2.4 Particle or Point Mass.
The smallest part of matter with zero dimension which can be described by its mass and position is defined as a particle.
If the size of a body is negligible in comparison to its range of motion then that body is known as a particle.
2.5 Distance and Displacement
(1) Distance: It is the actual path length covered by a moving particle in a given interval of time. (i) Its a scalar quantity. (ii) Dimension: [M0L1T0] (iii) Unit: metre (S.I.)
Kinematics / 69
(2) Displacement: Displacement is the change in position vector i.e., A vector joining initial to final position. (i) Displacement is a vector quantity (ii) Dimension: [M0L1T0] (iii) Unit: metre (S.I.)
(iv) If 1 2 3, , ........ nS S S S� � � �
are the displacements of a body then the total (net) displacement
is the vector sum of the individuals. 1 2 3 ........ nS S S S S= + + + +� � � � �
(3) Comparison between distance and displacement: (i) Distance ≥|Displacement|. (ii) For a moving particle distance can never be negative or zero while displacement can
be. i.e., Distance > 0 but Displacement > = or < 0 (iii) For motion between two points displacement is single valued while distance depends
on actual path and so can have many values. (iv) For a moving particle distance can never decrease with time while displacement
can. Decrease in displacement with time means body is moving towards the initial position.
(v) In general magnitude of displacement is not equal to distance. However, it can be so if the motion is along a straight line without change in direction.
2.6 Speed and Velocity.
(1) Speed: Rate of distance covered with time is called speed. (i) It is a scalar quantity having symbol υ . (ii) Dimension: [M0L1T–1] (iii) Unit: metre/second (S.I.), cm/second (C.G.S.) (iv) Types of speed:
(a) Uniform speed: When a particle covers equal distances in equal intervals of time, (no matter how small the intervals are) then it is said to be moving with uniform speed.
(b) Non-uniform (variable) speed: In non-uniform speed particle covers unequal distances in equal intervals of time.
(c) Average speed: The average speed of a particle for a given ‘Interval of time’ is defined as the ratio of distance travelled to the time taken.
Average speed Distance travelled
Time taken= ; av
sv
t
∆=
∆
� Time average speed: When particle moves with different uniform speed υ1, υ2, υ3 ... etc in different time intervals t1, t2, t3, ... etc respectively, its average speed over the total time of journey is given as
Total distance coveredTotal time elapsedavv = 1 2 3
1 2 3
............
d d d
t t t
+ + +=
+ + + = 1 1 2 2 3 3
1 2 3
............
t t t
t t t
υ + υ + υ +
+ + +
Special case: When particle moves with speed v1 upto half time of its total
motion and in rest time it is moving with speed v2 then 1 2
2av
v vv
+=
� Distance averaged speed: When a particle describes different distances d1, d2, d3, ...... with different time intervals t1, t2, t3, ...... with speeds v1, v2, v3, …..
70 / Academic Excellence in Physics
respectively then the speed of particle averaged over the total distance can be given as
Total distance covered
Total time elapsed 1 2 3
1 2 3
............
d d d
t t t
+ + +
+ + + 1 2 3
1 2 3
1 2 3
......
......
d d d
d d d
+ + +
+ + +υ υ υ
(d) Instantaneous speed: It is the speed of a particle at particular instant. When we say “speed”, it usually means instantaneous speed.
The instantaneous speed is average speed for infinitesimally small time interval (i.e., ∆t → 0). Thus
Instantaneous speed 0
limt
s dsv
t dt∆ →
∆= =
∆
(2) Velocity: Rate of change of position i.e. rate of displacement with time is called velocity. (i) It is a scalar quantity having symbol v. (ii) Dimension: [M0L1T–1] (iii) Unit: metre/second (S.I.), cm/second (C.G.S.) (iv) Types
(a) Uniform velocity: A particle is said to have uniform velocity, if magnitudes as well as direction of its velocity remains same and this is possible only when the particles moves in same straight line without reversing its direction.
(b) Non-uniform velocity: A particle is said to have non-uniform velocity, if either of magnitude or direction of velocity changes (or both changes).
(c) Average velocity: It is defined as the ratio of displacement to time taken by the body
Average velocity = DisplacementTime taken
; =av
rv
t
∆
∆
��
(d) Instantaneous velocity: Instantaneous velocity is defined as rate of change of position vector of particles with time at a certain instant of time.
Instantaneous velocity 0
limt
r drv
t dt→
∆= =
∆
� ��
(v) Comparison between instantaneous speed and instantaneous velocity (a) instantaneous velocity is always tangential to the path followed by the particle. (b) A particle may have constant instantaneous speed but variable instantaneous
velocity. (c) The magnitude of instantaneous velocity is equal to the instantaneous speed. (d) If a particle is moving with constant velocity then its average velocity and
instantaneous velocity are always equal. (e) If displacement is given as a function of time, then time derivative of
displacement will give velocity. (vi) Comparison between average speed and average velocity
(a) Average speed is scalar while average velocity is a vector both having same units (m/s) and dimensions [LT–1].
(b) Average speed or velocity depends on time interval over which it is defined. (c) For a given time interval average velocity is single valued while average speed
can have many values depending on path followed.
Kinematics / 71
(d) If after motion body comes back to its initial position then 0avv =�� (as 0r∆ =
� )
but 0avv >�
and finite as (∆s > 0).
(e) For a moving body average speed can never be negative or zero (unless t → ∞) while average velocity can be i.e. vav > 0 while av υ
� = or < 0.
2.7 Acceleration.
The time rate of change of velocity of an object is called acceleration of the object. (1) It is a vector quantity. It’s direction is same as that of change in velocity (Not of the
velocity) (2) There are three possible ways by which change in velocity may occur
When only direction of
velocity changes
When only magnitude of
velocity changes
When both magnitude and
direction of velocity
changes
Acceleration perpendicular to velocity
Acceleration parallel or anti-parallel to velocity
Acceleration has two components one is perpendicular to velocity and another parallel or anti-parallel to velocity
e.g. Uniform circular motion
e.g. Motion under gravity e.g. Projectile motion
(3) Dimension: [M0L1T–2] (4) Unit: metre/second2 (S.I.); cm/second2 (C.G.S.) (5) Types of acceleration:
(i) Uniform acceleration: A body is said to have uniform acceleration if magnitude and direction of the acceleration remains constant during particle motion. � If a particle is moving with uniform acceleration, this does not necessarily
imply that particle is moving in straight line. e.g. Projectile motion. (ii) Non-uniform acceleration: A body is said to have non-uniform acceleration, if
magnitude or direction or both, change during motion.
(iii) Average acceleration: 2 1a
v v va
t tυ
∆ −= =
∆ ∆
� � ��
�
The direction of average acceleration vector is the direction of the change in velocity
vector as v
at
∆=
∆
��
(iv) Instantaneous acceleration = 0
limt
v dva
t dt∆ →
∆= =
∆
� ��
(v) For a moving body there is no relation between the direction of instantaneous velocity and direction of acceleration. e.g. (a) In uniform circular motion θ = 90º always (b) In a projectile motion θ is
variable for every point of trajectory.
72 / Academic Excellence in Physics
(vi) By definition 2
2
dv d xa
dt dt= =� �
� As
dxv
dt
=
��
(vii) If velocity is given as a function of position, then by chain rule
asdv dv dx d dx
a v vdt dx dt dx dt
υ = = × = ⋅ =
(viii) If a particle is accelerated for a time t1 by acceleration a1 and for time t2 by
acceleration a2 then average acceleration is 1 1 2 2
1 2a
a t a ta
t tυ
+=
+
(ix) Acceleration can be positive, zero or negative. Positive acceleration means velocity increasing with time, zero acceleration means velocity is uniform constant while negative acceleration (retardation) means velocity is decreasing with time.
(x) For motion of a body under gravity, acceleration will be equal to “g”, where g is the acceleration due to gravity. Its normal value is 9.8 m/s2 or 980 cm/s2 or 32 feet/s2.
2.8 Position Time Graph
Various position – time graphs and their interpretation
θ = 0o so v = 0
i.e., line parallel to time axis represents that the particle is at rest.
θ = 90o so v = ∞
i.e., line perpendicular to time axis represents that particle is changing its position but time does not changes it means the particle possesses infinite velocity.
Practically this is not possible.
θ = constant so v = constant, a = 0
i.e., line with constant slope represents uniform velocity of the particle.
θ is increasing so v is increasing, a is positive.
i.e., line bending towards position axis represents increasing velocity of particle. It means the particle possesses acceleration.
θ is decreasing so v is decreasing, a is negative
i.e., line bending towards time axis represents decreasing velocity of the particle. It means the particle possesses retardation.
T
P
O
T
P
O
T
P
O
T
P
O
O T
P
Kinematics / 73
θ constant but > 90o so v will be constant but negative
i.e., line with negative slope represent that particle returns towards the point of reference. (negative displacement).
Straight line segments of different slopes represent that velocity of the body changes after certain interval of time.
This graph shows that at one instant the particle has two positions. Which is not possible.
The graph shows that particle coming towards origin initially and after that it is moving away from origin.
Note:
� If the graph is plotted between distance and time then it is always an increasing curve and it never comes back towards origin because distance never decrease with time.
� For two particles having displacement time graph with slopes θ1 and θ2 possesses
velocities v1 and v2 respectively then υ θ
=υ θ1 1
2 2
tan
tan
2.9 Velocity Time Graph.
The graph is plotted by taking time t along x-axis and velocity of the particle on y-axis. Distance and displacement: The area covered between the velocity time graph and time
axis gives the displacement and distance travelled by the body for a given time interval.
Then Total distance = Addition of modulus of different area. i.e. = υ∫| |s dt
Total displacement = Addition of different area considering their sign. i.e. = υ∫r dt
Acceleration: It is clear that slope of velocity-time graph represents the acceleration of the particle.
O T
P
θ
O T
P
S
C B A
O T
P
O T
P
74 / Academic Excellence in Physics
θ = 0, a = 0, v = constant
i.e., line parallel to time axis represents that the particle is moving with constant velocity.
θ = 90o, a = ∞, v = increasing
i.e., line perpendicular to time axis represents that the particle is increasing its velocity, but time does not change. It means the particle possesses infinite acceleration. Practically it is not possible.
θ =constant, so a = constant and v is increasing uniformly with time
i.e., line with constant slope represents uniform acceleration of the particle.
θ increasing so acceleration increasing
i.e., line bending towards velocity axis represent the increasing acceleration in the body.
θ decreasing so acceleration decreasing
i.e. line bending towards time axis represents the decreasing acceleration in the body
Positive constant acceleration because θ is constant and < 90o but initial velocity of the particle is negative.
Positive constant acceleration because θ is constant and < 90o but initial velocity of particle is positive.
Time O
Velo
city
Time O
Velo
city
Time O
Velo
city
Time O
Velo
city
Velo
city
O Time
O Time
Velo
city
V
elo
city
O Time
Various velocity – time graphs and their interpretation
Kinematics / 75
Negative constant acceleration because θ is constant and > 90o but initial velocity of the particle is positive.
Negative constant acceleration because θ is constant and > 90o but initial velocity of the particle is zero.
Negative constant acceleration because θ is constant and > 90o but initial velocity of the particle is negative.
2.10 Equations of Kinematics.
These are the various relations between u, v, a, t and s for the moving particle where the notations are used as:
u = Initial velocity of the particle at time t = 0 sec v = Final velocity at time t sec a = Acceleration of the particle s = Distance travelled in time t sec sn = Distance travelled by the body in nth sec (1) When particle moves with constant acceleration
(i) Acceleration is said to be constant when both the magnitude and direction of acceleration remain constant.
(ii) There will be one dimensional motion if initial velocity and acceleration are parallel or anti-parallel to each other.
(iii) Equations of motion in scalar from Equation of motion in vector from υ = +u at v u at= +
� � �
212
s ut at= + 212
s ut at= +� � �
2 2 2u asυ = + . . 2 .v v u u a s− =� � � � � �
2
u vs t
+ =
1
( )2
s u v t= +� � �
(2 1)2n
as u n= + − (2 1)
2n
as u n= + −
�� �
(2) Important points for uniformly accelerated motion (i) If a body starts from rest and moves with uniform acceleration then distance
covered by the body in t sec is proportional to t2 (i.e. ∝ 2s t ). So the ratio of distance covered in 1 sec, 2 sec and 3 sec is 12: 22:32 or 1: 4: 9.
(ii) If a body starts from rest and moves with uniform acceleration then distance covered by the body in nth sec is proportional to (2n –1) (i.e. sn ∝ (2n – 1). So the ratio of distance covered in I sec, II sec and III sec is 1: 3: 5.
O Time
Velo
city
O Time
Velo
city
O Time
Velo
city
76 / Academic Excellence in Physics
(iii) A body moving with a velocity u is stopped by application of brakes after covering a distance s. If the same body moves with velocity nu and same braking force is applied on it then it will come to rest after covering a distance of n2s.
2.11 Motion of Body Under Gravity (Free Fall).
Acceleration produced in the body by the force of gravity, is called acceleration due to gravity. It is represented by the symbol g.
In the absence of air resistance, it is found that all bodies fall with the same acceleration near the surface of the earth. This motion of a body falling towards the earth from a small altitude (h << R) is called free fall.
An ideal one-dimensional motion under gravity in which air resistance and the small changes in acceleration with height are neglected.
(1) If a body dropped from some height (initial velocity zero) (i) Equation of motion: Taking initial position as origin and direction of motion (i.e.,
downward direction) as a positive, here we have u = 0 [As body starts from rest] a = +g [As acceleration is in the direction of motion] v = gt …(i)
212
h gt= …(ii)
υ2 = 2gh …(iii)
(2 1)2n
gh n= − ...(iv)
(ii) Graph of distance velocity and acceleration with respect to time:
(2) If a body is projected vertically downward with some initial velocity Equation of motion: υ = u + gt
212
h ut g t= +
υ 2 = u2 + 2gh
(2 1)2n
gh u n= + −
(3) If a body is projected vertically upward (i) if the body is projected with velocity u and after time t it reaches up to height h then
υ = u – gt; 212
h ut g t= − ; υz = u2 – 2gh; (2 1)2n
gh u n= − −
s
t
a
g
t
v
θ
t
tanθ = g
Kinematics / 77
(ii) For maximum height v = 0 So from above equation u = gt,
212
h gt=
and u2 = 2gh (iii) Graph of distance, velocity and acceleration with respect to time (for maximum
height):
It is clear that both quantities do not depend upon the mass of the body or we can say
that in absence of air resistance, all bodies fall on the surface of the earth with the same rate.
(4) The motion is independent of the mass of the body, as in any equation of motion, mass is not involved. That is why a heavy and light body when released from the same height, reach the ground simultaneously and with same velocity i.e., (2 / )t h g= and 2v gh= .
(6) In case of motion under gravity time taken to go up is equal to the time taken to fall down through the same distance.
(7) In case of motion under gravity, the speed with which a body is projected up is equal to the speed with which it comes back to the point of projection.
(8) A ball is dropped from a building of height h and it reaches after t seconds on earth. From the same building if two ball are thrown (one upwards and other downwards) with the same velocity u and they reach the earth surface after t1 and t2 seconds respectively then
1 2t t t=
(9) A body is thrown vertically upwards. If air resistance is to be taken into account, then the time of ascent is less than the time of descent. t2 > t1
2.12 Motion with Variable Acceleration
(i) If acceleration is a function of time
a = f(t), then 0
( )t
v u f t dt= + ∫ and ( )( )s ut f t dt dt= + ∫ ∫
(ii) If acceleration is a function of distance
a = f(x) then 0
2 2 2 ( )x
xv u f x dx= + ∫
(iii) If acceleration is a function of velocity
a = f (v) then ( )
v
u
dvt
f v= ∫ and 0 ( )
v
u
vdvx x
f v= + ∫
s
t
(u/g)
(u2/2g)
(u/g) (2u/g) t
v
O
– v
+
–
t
a
O
g
+
– a
78 / Academic Excellence in Physics
The motion of an object is called two dimensional, if two of the three co-ordinates are required to specify the position of the object in space changes w.r.t time.
In such a motion, the object moves in a plane. For example, a billiard ball moving over the billiard table, an insect crawling over the floor of a room, earth revolving around the sun etc.
Two special cases of motion in two dimension are: 1. Projectile motion 2. Circular motion
ROJECTILE MOTION
2.13 Introduction.
If the force acting on a particle is oblique with initial velocity then the motion of particle is called projectile motion.
2.14 Projectile
A body which is in flight through the atmosphere but is not being propelled by any fuel is called projectile.
2.15 Assumptions of Projectile Motion.
(1) There is no resistance due to air. (2) The effect due to curvature of earth is negligible. (3) The effect due to rotation of earth is negligible. (4) For all points of the trajectory, the acceleration due to gravity ‘g’ is constant in
magnitude and direction.
2.16 Principles of Physical Independence of Motions.
(1) The motion of a projectile is a two-dimensional motion. So, it can be discussed in two parts. Horizontal motion and vertical motion. These two motions take place independent of each other. This is called the principle of physical independence of motions.
(2) The velocity of the particle can be resolved into two mutually perpendicular components. Horizontal component and vertical component.
(3) The horizontal component remains unchanged throughout the flight. The force of gravity continuously affects the vertical component.
(4) The horizontal motion is a uniform motion and the vertical motion is a uniformly accelerated retarded motion.
2.17 Types of Projectile Motion
(1) Oblique projectile motion (2) Horizontal projectile motion (3) Projectile motion on an inclined plane
2.18 Oblique Projectile.
In projectile motion, horizontal component of velocity (u cosθ), acceleration (g) and mechanical energy remains constant while, speed, velocity, vertical component of velocity (u sin θ), momentum, kinetic energy and potential energy all changes. Velocity, and KE are maximum at the point of projection while minimum (but not zero) at highest point.
(1) Equation of trajectory: A projectile thrown with velocity u at an angle θ with the horizontal.
θ
X
Y
O
u u sin θ
u cos θ
y
x P
Kinematics / 79
vx
The velocity u can be resolved into two rectangular components u cos θ component along
X–axis and u sin θ component along Y–axis. 2
2 2
1tan
2 cosgx
y xu
= θ −θ
Note:
� Equation of oblique projectile also can be written as
tan 1x
y xR
= θ − (where R = horizontal range )
(2) Displacement of projectile ( )r�
: Let the particle acquires a position P having the coordinates (x, y) just after time t from the instant of projection. The corresponding position vector of the particle at time t is
�r
as shown in the figure.
ˆ ˆr xi yj= +� ….(i)
The horizontal distance covered during time t is given as
cosxx v t x u t= ⇒ = θ ….(ii)
The vertical velocity of the particle at time t is given as 2sin 1 / 2y u t gt= θ − ….(iii)
and 1tan ( / )y x−φ = Note:
� The angle of elevation φ of the highest point of the projectile and the angle of
projection θ are related to each other as 1
tan tan2
φ = θ
(3) Instantaneous velocity v: In projectile motion, vertical component of velocity changes but horizontal component of velocity remains always constant.
Let vi be the instantaneous velocity of projectile at time t direction of this velocity is along the tangent to the trajectory at point P.
ˆi x yv v i v j= +� ⇒ 2 2
i x yv v v= +
Direction of instantaneous velocity sin
tancos
y
x
v u gt
v u
θ −α = =
θ
(7) Time of flight: The total time taken by the projectile to go up and come down to the same level from which it was projected is called time of flight.
For vertical upward motion 0 = u sin θ – gt ⇒ t = (u sin θ/g)
Time of flight 2 sin2
uT t
g
θ= =
(8) Horizontal range: It is the horizontal distance travelled by a body during the time of flight.
So by using second equation of motion
R = u cos θ × T = u cos θ × (2u sin θ/g) 2 sin 2u
g
θ=
φ θ
v�
α xv
�
yv�
P (x, y)
x O
y r�
X
Y
vi
80 / Academic Excellence in Physics
2 sin 2u
Rg
θ=
If angle of projection is changed from θ to θ′ = (90 – θ) then range remains unchanged.
These angles are called complementary angles of projection.
(iv) Maximum range: For range to be maximum 0dR
d=
θ ⇒
2 sin 20
d u
d g
θ= θ
a projectile will have maximum range when it is projected at an angle of 45o to the horizontal and the maximum range will be (u2/g). When the range is maximum, the height H reached by the projectile
2 2 2 2 2
maxsin sin 452 2 4 4
u u u RH
g g g
θ= = = =
(v) Relation between horizontal range and maximum height: R = 4H cot θ
If R = 4H then θ = tan–1(1) or θ = 45°. (9) Maximum height: It is the maximum height from the point of projection, a projectile
can reach. So, by using v2 = u2 + 2as 0 = (u sin θ)2 – 2gH
2 2sin
2u
Hg
θ=
(i) 2
max 2u
Hg
= (when sin2θ = max = 1 i.e., θ = 90o)
i.e., for maximum height body should be projected vertically upward. (10) Motion of a projectile as observed from another projectile is a straight line.
2.19 Horizontal Projectile
A body be projected horizontally from a certain height ‘y’ vertically above the ground with initial velocity u. If friction is considered to be absent, then there is no other horizontal force which can affect the horizontal motion. The horizontal velocity therefore remains constant.
(4) Time of flight: If a body is projected horizontally from a height h with velocity u and
time taken by the body to reach the ground is T, then 2h
Tg
=
(5) Horizontal range: Let R is the horizontal distance travelled by the body 2h
R ug
=
(6) If projectiles A and B are projected horizontally with different initial velocity from same height and third particle C is dropped from same point then (i) All three particles will take equal time to reach the ground. (ii) Their net velocity would be different but all three particle possess same vertical
component of velocity. (iii) The trajectory of projectiles A and B will be straight line w.r.t. particle C.
(7) If various particles thrown with same initial velocity but indifferent direction then
Kinematics / 81
(i) They strike the ground with same speed at different times irrespective of their initial direction of velocities.
(ii) Time would be least for particle which was thrown vertically downward. (iii) Time would be maximum for particle A which was thrown vertically upward.
2.20 Projectile Motion on an Inclined Plane.
Let a particle be projected up with a speed u from an inclined plane which makes an angle α with the horizontal velocity of projection makes an angle θ with the inclined plane.
We have taken reference x-axis in the direction of plane. Hence the component of initial velocity parallel
and perpendicular to the plane are equal to u cos θ and u sin θ respectively i.e. u|| cos θ and u⊥ = u sin θ.
The component of g along the plane is g sin α and perpendicular to the plane is αcosg as shown in the figure i.e. a|| = –g sin α and a⊥ = g cos α.
Therefore the particle decelerates at a rate of g sin α as it moves from O to P.
(1) Time of flight: We know for oblique projectile motion 2 sinuT
g
θ= or we can say 2u
Ta
⊥
⊥
=
∴ Time of flight on an inclined plane 2 sincos
uT
g
θ=
α
(2) Maximum height: We know for oblique projectile motion 2 2sin
2u
Hg
θ= or we can say
2
2u
Ha⊥
⊥
=
∴ Maximum height on an inclined plane 2 2sin
2 cosu
Hg
θ=
α
(3) Horizontal range: 2
2
2 sin cos( )cos
uR
g
θ θ + α=
α
(i) Maximum range occurs when 4 2π α
θ = −
(ii) The maximum range along the inclined plane when the projectile is thrown upwards
is given by 2
max (1 sin )u
Rg
=+ α
(iii) The maximum range along the inclined plane when the projectile is thrown
downwards is given by 2
max (1 sin )u
Rg
=− α
CIRCULAR MOTION
Circular motion is another example of motion in two dimensions. To create circular motion in a body it must be given some initial velocity and a force must then act on the body which is always directed at right angles to instantaneous velocity.
θ
O α
Y
X u�
t =0
t =T
g�
ay= g cos α
ax=–g sin α
P
82 / Academic Excellence in Physics
Circular motion can be classified into two types – Uniform circular motion and non-uniform circular motion.
2.21 Variables of Circular Motion.
(1) Displacement and distance: When particle moves in a circular path describing an angle θ during time t (as shown in the figure) from the position A to the position B, we see that the magnitude of the position vector r
� (that is equal to the radius of the circle) remains constant. i.e., 1 2r r r= =
� � and the direction of the position vector changes from
time to time.
(i) Displacement: The change of position vector or the displacement r∆
� of the particle from position A to the position B is given by referring the figure.
2 1r r r∆ = −� � �
2 sin2
r rθ
∆ =
(ii) Distance: The distanced covered by the particle during the time t is given as d = length of the arc AB = r θ
(2) Angular displacement (θ): The angle turned by a body moving on a circle from some reference line is called angular displacement. (i) Dimension = [M0L0T0] (as θ = arc/radius). (ii) Units = Radian or Degree. It is some times also
specified in terms of fraction or multiple of revolution. (iii) 2πrad = 360° = 1 Revolution (iv) Angular displacement is a axial vector quantity.
Its direction depends upon the sense of rotation of the object and can be given by Right Hand Rule; which states that if the curvature of the fingers of right hand represents the sense of rotation of the object, then the thumb, held perpendicular to the curvature of the fingers, represents the direction of angular displacement vector.
(v) Relation between linear displacement and angular displacement s r= θ ×� � �
or s = rθ
(3) Angular velocity (ω): Angular velocity of an object in circular motion is defined as the time rate of change of its angular displacement.
(i) Angular velocity ω =0
angle tracedtime taken t
dLt
t dt∆ →
∆θ θ= =
∆
∴ d
dt
θω =
(ii) Dimension: [M0L0T–1] (iii) Units: Radians per second (rad.s–1) or Degree per second. (iv) Angular velocity is an axial vector. Its direction is the same as that of ∆θ. (v) Relation between angular velocity and linear velocity = ω ×
�� �v r
O θ
1r�
2r�
A
B
1v�
2v�
1r�
2r�
O
r�
∆
A
B
θ
θ
r O
S
Kinematics / 83
(vi) For uniform circular motion ω remains constant where as for non-uniform motion ω varies with respect to time.
(4) Change in velocity: We want to know the magnitude and direction of the change in velocity of the particle which is performing uniform circular motion as it moves from A to B during time t as shown in figure. The change in velocity vector is given as
2 1v v v∆ = −� �
∆v 2 sin2
vθ
=
� Relation between linear velocity and angular velocity. In vector form v r= ω×
� � � (5) Time period (T): In circular motion, the time period is defined as the time taken by the
object to complete one revolution on its circular path. (6) Frequency (n): In circular motion, the frequency is defined as the number of revolutions
completed by the object on its circular path in a unit time. (i) Units: s–1 or hertz (Hz). (ii) Dimension: [M0L0T–1]
Note:
� Relation between time period and frequency: ∴ T = 1/n
� Relation between angular velocity, frequency and time period:
2
2 nT
πω = = π
(7) Angular acceleration (α): Angular acceleration of an object in circular motion is defined as the time rate of change of its angular velocity.
(i) 2
20t
d dLt
t dt dt∆ →
∆ω ω θα = = =
∆
(ii) Units: rad. s–2 (iii) Dimension: [M0L0T–2] (iv) Relation between linear acceleration and angular acceleration a r= α ×
� � �
(v) For uniform circular motion since ω is constant so 0d
dt
ωα = =
(vi) For non-uniform circular motion 0α ≠ .
2.22 Centripetal Acceleration
(1) Acceleration acting on the object undergoing uniform circular motion is called centripetal acceleration.
(2) It always acts on the object along the radius towards the centre of the circular path.
(3) Magnitude of centripetal acceleration2 2
2 22
44
va r n r r
r T
π= = ω = π =
(4) Direction of centripetal acceleration: It is always the same as that of v∆� .
2.23 Centripetal Force.
According to Newton's first law of motion, whenever a body moves in a straight line with uniform velocity, no force is required to maintain this velocity. But when a body moves along a circular
F
v v
v v
F F
F
84 / Academic Excellence in Physics
path with uniform speed, its direction changes continuously i.e. velocity keeps on changing on account of a change in direction. According to Newton's second law of motion, a change in the direction of motion of the body can take place only if some external force acts on the body.
Due to inertia, at every point of the circular path; the body tends to move along the tangent to the circular path at that point (in figure). Since every body has directional inertia, a velocity cannot change by itself and as such we have to apply a force. But this force should be such that it changes the direction of velocity and not its magnitude. This is possible only if the force acts perpendicular to the direction of velocity. Because the velocity is along the tangent, this force must be along the radius (because the radius of a circle at any point is perpendicular to the tangent at that point). Further, as this force is to move the body in a circular path, it must acts towards the centre. This centre-seeking force is called the centripetal force.
Hence, centripetal force is that force which is required to move a body in a circular path with uniform speed. The force acts on the body along the radius and towards centre.
(1) Formulae for centripetal force: 2 2
2 2 22
44
mv m rF m r m n r
r T
π= = ω = π =
(2) Centripetal force in different situation
Situation Centripetal Force
A particle tied to a string and whirled in a horizontal circle.
Tension in the string.
Vehicle taking a turn on a level road. Frictional force exerted by the road on the tyres.
A vehicle on a speed breaker. Weight of the body or a component of weight. Revolution of earth around the sun Gravitational force exerted by the sun. Electron revolving around the nucleus in an atom.
Coulomb attraction exerted by the protons in the nucleus.
A charged particle describing a circular path in a magnetic field.
Magnetic force exerted by the agent that sets up the magnetic field.
2.24 Centrifugal Force
It is an imaginary force due to incorporated effects of inertia. Centrifugal force is a fictitious force which has significance only in a rotating frame of reference.
2.25 Work done by Centripetal Force
The work done by centripetal force is always zero as it is perpendicular to velocity and hence instantaneous displacement.
Example: (i) When an electron revolve around the nucleus in hydrogen atom in a particular orbit, it neither absorb nor emit any energy means its energy remains constant.
(ii) When a satellite established once in a orbit around the earth and it starts revolving with particular speed, then no fuel is required for its circular motion.
2.26 Skidding of Vehicle on a Level Road.
When a vehicle turns on a circular path it requires centripetal force.
µmg mω2r
Kinematics / 85
If friction provides this centripetal force then vehicle can move in circular path safely if Friction force ≥ Required centripetal force
2mv
mgr
µ ≥
∴ safev rg≤ µ
This is the maximum speed by which vehicle can turn in a circular path of radius r, where coefficient of friction between the road and tyre is µ.
2.27 Skidding of Object on a Rotating Platform.
On a rotating platform, to avoid the skidding of an object (mass m) placed at a distance r from axis of rotation, the centripetal force should be provided by force of friction.
Centripetal force = Force of friction mω2r = µmg
∴ max ( / ) ,g rω = µ
Hence maximum angular velocity of rotation of the platform is ( / ) ,g rµ so that object will not skid on it.
2.28 Bending of a Cyclist
A cyclist provides himself the necessary centripetal force by leaning inward on a horizontal track, while going round a curve. Consider a cyclist of weight mg taking a turn of radius r with velocity v. In order to provide the necessary centripetal force, the cyclist leans through angle θ inwards as shown in figure.
2
sinmv
Rr
θ = …..(i)
and R cos θ = mg …..(ii) Dividing equation (i) by (ii), we have
2
tanv
rgθ = …..…(iii)
Note:
� For the same reasons, an ice skater or an aeroplane has to bend inwards, while taking a turn.
2.29 Banking of a Road.
For getting a centripetal force cyclist bend towards the centre of circular path but it is not possible in case of four wheelers.
Therefore, outer bed of the road is raised so that a vehicle moving on it gets automatically inclined towards the centre.
mg
R cosθ
R sinθ mv2/r
θ
R
86 / Academic Excellence in Physics
2
tanv
r gθ = ...... (iii)
or 2
tanr v
g rg
ω ωθ = = ...... (iv) [As v = rω ]
If l = width of the road, h = height of the outer edge from the ground level then from the figure (B)
tanh h
x lθ = = .......(v) [since θ is very small]
� Maximum safe speed on a banked frictional road ( tan )
1 tanrg
vµ + θ
=− µ θ
2.30 Overturning of Vehicle.
When a car moves in a circular path with speed more than maximum speed then it overturns and it’s inner wheel leaves the ground first
Weight of the car = mg Speed of the car = v Radius of the circular path = r Distance between the centre of wheels of the car = 2a Height of the centre of gravity (G) of the car from the road level = h
The maximum speed of a car without overturning on a flat road is given by grav
h= .
2.31 Non-Uniform Circular Motion.
If the speed of the particle in a horizontal circular motion changes with respect to time, then its motion is said to be non-uniform circular motion.
using rυ = ω×� � � …..(i)
the resultant acceleration of the particle at P has two component accelerations (1) Tangential acceleration: ta r= α ×
� � �
It acts along the tangent to the circular path at P in the plane of circular path. (2) Centripetal (Radial) acceleration:
ca v= ω ×�� �
It is also called centripetal acceleration of the particle at P. It acts along the radius of the particle at P. The magnitude of centripetal acceleration is given by | | | | sin90ca = ω×υ = ωυ °
� � � = 2 2( ) /r r rωυ= ω ω = ω = υ
R cos θ
mg
θ
R
R sin θ
θ
θ
Fig. (A)
θ x
l
h
Fig. (B)
Kinematics / 87
� Here at governs the magnitude of v� while ca
� its direction of motion.
2.32 Equations of Circular Motion.
For accelerated
motion
For retarded
motion
ω2 = ω1 + αt ω2 = ω1 + αt
21
12
t tθ = ω + α 21
12
t tθ = ω − α
2 1
2 2 2ω = ω + α θ 2 1
2 2 2ω = ω − α θ
1 (2 1)2n nα
θ = ω + − 1 (2 1)2n nα
θ = ω − −
2.33 Motion in Vertical Circle.
This is an example of non-uniform circular motion. In this motion body is under the influence of gravity of earth.
(1) Velocity at any point on vertical loop: If u is the initial velocity imparted to body at lowest point then. Velocity of body at height h is given by
2 22 2 (1 cos )v u gh u gl= − = − − θ
where l in the length of the string
(2) Tension at any point on vertical loop: Tension at general point P, 2
cosmv
T mgl
= θ +
(3) Various conditions for vertical motion:
Velocity at lowest
point
Condition
5Au gl> Tension in the string will not be zero at any of the point and body will continue the circular motion.
5 ,Au gl= Tension at highest point C will be zero and body will just complete the circle.
2 5 ,Agl u gl< < Particle will not follow circular motion. Tension in string become zero somewhere between points B and C whereas velocity remain positive. Particle leaves circular path and follow parabolic trajectory.
2Au gl= Both velocity and tension in the string becomes zero between A and B and particle will oscillate along semi-circular path.
2Au gl< velocity of particle becomes zero between A and B but tension will not be zero and the particle will oscillate about the point A.
θ
O l
P v
h
A u
D
C
B
Where ω1 = Initial angular velocity of particle ω2 = Final angular velocity of particle α = Angular acceleration of particle θ = Angle covered by the particle in time t θn = Angle covered by the particle in nth second
88 / Academic Excellence in Physics
(6) Various quantities for a critical condition in a vertical loop at different
positions:
Quantity Point
A
Point B Point C Point D Point P
Linear velocity (v) 5gl 3gl gl 3gl (3 2cos )gl + θ
Angular velocity (ω) 5g
l 3g
l g
l 3g
l (3 2cos )
g
l+ θ
Tension in String (T)
6 mg 3 mg 0 3 mg 3mg (1 + cos θ)
Kinetic Energy (KE) 52
mgl 32
mgl 12
mgl 32
mgl 2
5 0mu
mgl
− =
Potential Energy (PE)
0 mgl 2 mgl mgl Mgl(1 – cos θ)
Total Energy (TE) 52
mgl 52
mgl 52
mgl 52
mgl 52
mgl
(7) Motion of a block on frictionless
hemisphere: A small block of mass m slides down from the top of a frictionless hemisphere of radius r. The component of the force of gravity (mg cos θ) provides required centripetal force but at point B it's circular motion ceases and the block lose contact with the surface of the sphere.
For point B, by equating the forces, 2
cosmv
mgr
θ = .....(i)
by law of conservation of energy
Total energy at point A = Total energy at point B the block lose contact at the height of 23
r
from the ground.
h
mg
(r – h) B
θ θ r
A
Newton’s Laws of Motion and Friction / 89
3. Newton’s Laws of Motion
3.1 Point Mass.
3.2 Inertia
(1) Inherent property of all the bodies by virtue of which they cannot change their state of rest or uniform motion along a straight line by their own is called inertia.
(2) Two bodies of equal mass possess same inertia because it is a factor of mass only.
3.3 Linear Momentum.
(1) Linear momentum of a body is the quantity of motion contained in the body. (2) It is measured as the product of the mass of the body and its velocity i.e., Momentum = mass
× velocity.
If a body of mass m is moving with velocity v��
then its linear momentum p��
is given by
p mv=�� ��
(3) It is a vector quantity and it’s direction is the same as the direction of velocity of the body. (4) Units: kg-m/sec [S.I.], g-cm/sec [C.G.S.] (5) Dimension: [MLT–1]
3.4 Newton’s First Law
A body continue to be in its state of rest or of uniform motion along a straight line, unless it is acted upon by some external force to change the state.
(1) If no net force acts on a body, then the velocity of the body cannot change i.e. the body cannot accelerate.
(2) Newton’s first law defines inertia and is rightly called the law of inertia. Inertia are of three types:
Inertia of rest, Inertia of motion, Inertia of direction
3.5 Newton’s Second Law.
(1) The rate of change of linear momentum of a body is directly proportional to the external force applied on the body and this change takes place always in the direction of the applied force.
(2) If a body of mass m, moves with velocity v�
then its linear momentum can be given by p mv=� �
and if force F→
is applied on a body, then
or dp
Fdt
=�
�
(K = 1 in C.G.S. and S.I. units)
∴ F ma=� �
( Force = mass × acceleration)
3.6 Force
(1) Force is an external effect in the form of a push or pulls which: (i) Produces or tries to produce motion in a body at rest. (ii) Stops or tries to stop a moving body.
90 / Academic Excellence in Physics
(iii) Changes or tries to change the direction of motion of the body.
(2) Dimension: Force = mass × acceleration
2 2[ ] [ ][ ] [ ]F M LT MLT− −= = (3) Units: Absolute units: (i) Newton (S.I.) (ii) Dyne (C.G.S) Gravitational units: (i) Kilogram-force (M.K.S.) (ii) Gram-force (C.G.S)
(4) F ma=� �
formula is valid only if force is changing the state of rest or motion and the mass of the body is constant and finite.
(5) If m is not constant ( )d dv dm
F mv m vdt dt dt
= = +�
� � �
(6) No force is required to move a body uniformly along a straight line.
F ma=���
0F∴ =���
(As a = 0) (7) When force is written without direction then positive force means repulsive while negative
force means attractive. (8) Out of so many natural forces nuclear force is strongest while gravitational force weakest. (9) Central force: If a position dependent force is always directed towards or away from a fixed
point it is said to be central otherwise non-central. (10) Conservative or non conservative force: If under the action of a force the work done in a
round trip is zero or the work is path independent, the force is said to be conservative otherwise non conservative.
Example: Conservative force: Gravitational force, electric force, elastic force. Non conservative force: Frictional force, viscous force. (11) Common forces in mechanics:
(i) Weight: Weight of an object is the force with which earth attracts it.(W=mg) (ii) Reaction or Normal force: When a body is placed on a rigid surface, the body experiences
a force which is perpendicular to the surfaces in contact. Then force is called ‘Normal force’ or ‘Reaction’.
(iii) Tension: The force exerted by the end of taut string, rope or chain against pulling (applied) force is called the tension. The direction of tension is so as to pull the body.
(iv) Spring force: Every spring resists any attempt to change its length. This resistive force increases with change in length. Spring force is given by KxF −= ; where x is the change in length and K is the spring constant (unit N/m).
3.7 Equilibrium of Concurrent Force.
(1) If all the forces working on a body are acting on the same point, then they are said to be concurrent.
(2) A body, under the action of concurrent forces, is said to be in equilibrium, when there is no change in the state of rest or of uniform motion along a straight line.
(3) The condition for the equilibrium of a body is that the vector sum of all the forces acting on the body must be zero.
3.8 Newton’s Third Law.
To every action, there is always an equal (in magnitude) and opposite (in direction) reaction.
If ABF���
= force exerted on body A by body B (Action) and BAF���
= force exerted on body B by body A (Reaction).
Newton’s Laws of Motion and Friction / 91
Then according to Newton’s third law of motion AB BAF F= −��� ���
Example: (i) A book lying on a table exerts a force on the table which is equal to the weight of the
book. This is the force of action. (ii) Swimming is possible due to third law of motion. (iii) When a gun is fired, the bullet moves forward (action). The gun recoils backward
(reaction)
3.9 Frame of Reference.
(1) A frame in which an observer is situated and makes his observations is known as his ‘Frame of reference’.
It is associated with a co-ordinate system. (2) Frame of reference are of two types: (i) Inertial frame of reference (ii) Non-inertial frame of
reference. (i) Inertial frame of reference:
(a) A frame of reference which is at rest or which is moving with a uniform velocity along a straight line is called an inertial frame of reference.
(b) In inertial frame of reference Newton’s laws of motion holds good. (c) Ideally no inertial frame exist in universe. For practical purpose a frame of
reference may be considered as inertial if it’s acceleration is negligible with respect to the acceleration of the object to be observed.
Example: The lift at rest, lift moving (up or down) with constant velocity, (ii) Non inertial frame of reference:
(a) Accelerated frame of references are called non-inertial frame of reference. (b) Newton’s laws of motion are not applicable in non-inertial frame of reference. Example: Car moving in uniform circular motion, lift which is moving upward or
downward with some acceleration, plane which is taking off.
3.10 Impulse
(1) When a large force works on a body for very small time interval, it is called impulsive force. An impulsive force does not remain constant, but changes first from zero to maximum and
then from maximum to zero. In such case we measure the total effect of force. (2) Impulse of a force is a measure of total effect of force.
(3) 2
1
t
tI F dt= ∫�� ���
.
(4) Impulse is a vector quantity and its direction is same as that of force. (5) Dimension: [MLT–1] (6) Units: Newton-second or Kg-m-s–1 (S.I.) and Dyne-second or gm-cm s–1 (C.G.S.)
(7) Force-time graph: Impulse is equal to the area under F-t
curve.
=I Area between curve and time axis 12
F t=
(8) If Fav is the average magnitude of the force then
Fo
rce
Time t
F
92 / Academic Excellence in Physics
2 2
1 1
t t
av avt t
I F dt F dt F t= = = ∆∫ ∫
(9) From Newton’s second law d p
Fdt
→
=
��
or 2 2
1 1
t p
t pF dt d p=∫ ∫��� ��
⇒ 2 1I p p p= − = ∆� �� �� ���
i.e. The impulse of a force is equal to the change in momentum. This statement is known as Impulse momentum theorem. (10) Examples: Hitting, kicking, catching, jumping, diving, collision etc.
In all these cases an impulse acts. .avI F dt F t p= = ∆ = ∆ =∫ constant
So if time of contact ∆t is increased, average force is decreased (or diluted) and vice-versa. (i) In catching a ball a player by drawing his hands backwards increases the time of contact
and so, lesser force acts on his hands and his hands are saved from getting hurt. (ii) China wares are wrapped in straw or paper before packing.
3.11 Law of Conservation of Linear Momentum.
If no external force acts on a system (called isolated) of constant mass, the total momentum of the system remains constant with time.
(1) According to this law for a system of particles dt
pdF =
In the absence of external force 0=F�
then =p�
constant
i.e., =+++= ....321pppp constant.
(2) Law of conservation of linear momentum is independent of frame of reference though linear momentum depends on frame of reference.
(3) Practical applications of the law of conservation of linear momentum
(i) When a man jumps out of a boat on the shore, the boat is pushed slightly away from the shore.
(ii) A person left on a frictionless surface can get away from it by blowing air out of his mouth or by throwing some object in a direction opposite to the direction in which he wants to move.
(iii) Recoiling of a gun: For bullet and gun system, the force exerted by trigger will be internal so the momentum of the system remains unaffected.
(iv) Rocket propulsion: The initial momentum of the rocket on its launching pad is zero. When it is fired from the launching pad, the exhaust gases rush downward at a high speed and to conserve momentum, the rocket moves upwards.
Let m0 = initial mass of rocket, m = mass of rocket at any instant ‘t’ (instantaneous mass) mr = residual mass of empty container of the rocket u = velocity of exhaust gases, v = velocity of rocket at any instant ‘t’ (instantaneous velocity)
dm
dt = rate of change of mass of rocket = rate of fuel consumption
Newton’s Laws of Motion and Friction / 93
= rate of ejection of the fuel.
(a) Thrust on the rocket: dm
F u mgdt
= − −
Here negative sign indicates that direction of thrust is opposite to the direction of escaping gases.
dm
F udt
= − (if effect of gravity is neglected)
(b) Acceleration of the rocket: u dm
a gm dt
= − and if effect of gravity is neglected u dm
am dt
=
(c) Instantaneous velocity of the rocket: 0log e
mv u gt
m
= −
and if effect of gravity is
neglected 0log e
mv u
m
=
0102.303 log
mu
m
=
(d) Burnt out speed of the rocket: 0max logb e
r
mv v u
m
= =
The speed attained by the rocket when the complete fuel gets burnt is called burnt out speed of the rocket. It is the maximum speed acquired by the rocket.
3.12 Free Body Diagram
In this diagram the object of interest is isolated from its surroundings and the interactions between the object and the surroundings are represented in terms of forces.
3.13 Apparent Weight of a Body in a Lift.
When a body of mass m is placed on a weighing machine which is placed in a lift, then actual weight of the body is mg.
This acts on a weighing machine which offers a reaction R given by the reading of weighing machine. The reaction exerted by the surface of contact on the body is the apparent weight of the body.
Condition Figure Velocity Acceleration Reaction Conclusion
Lift is at rest
v = 0 a = 0 R – mg = 0 ∴ R = mg
Apparent weight = Actual weight
Lift moving upward or downward with constant velocity
v = constant
a = 0 R – mg = 0 ∴ R = mg
Apparent weight = Actual weight
Spring Balance
R
mg
LIFT
Spring Balance
R
mg
LIF
94 / Academic Excellence in Physics
Lift accelerating upward at the rate of 'a’
v = variable
a < g
R – mg = ma ∴R = m(g + a)
Apparent weight > Actual weight
Lift accelerating upward at the rate of ‘g’
v = variable
a = g R – mg = mg R = 2mg
Apparent weight = 2 Actual weight
Lift accelerating downward at the rate of ‘a’
v = variable
a < g
mg – R = ma ∴ R = m(g – a)
Apparent weight < Actual weight
Lift accelerating downward at the rate of ‘g’
v = variable
a = g mg – R = mg R = 0
Apparent weight = Zero (weightlessness)
Lift accelerating downward at the rate of a(>g)
v = variable
a > g
mg – R = ma R = mg – ma R = – ve
Apparent weight negative means the body will rise from the floor of the lift and stick to the ceiling of the lift.
3.14 Acceleration of Block on Horizontal Smooth Surface.
(1) When a pull is acting at an angle (θ) to the horizontal (upward)
R + F sin θ = mg
⇒ R = mg – F sinθ and F cosθ = ma
∴ m
Fa
θ=
cos
a
Spring Balance
R
mg
LIF
g
Spring Balance
R
mg
LIF
a
Spring Balance
R
mg
LIFT
g
Spring Balance
R
mg
LIFT a >
g
Spring Balance
R
mg
LIFT
mg
R
m F cosθ
F
F sinθ
θ
Newton’s Laws of Motion and Friction / 95
3.15 Acceleration of Block on Smooth Inclined Plane
(1) When inclined plane is at rest Normal reaction R = mg cosθ Force along a inclined plane F = mg sinθ
ma = mg sinθ ∴ a = g sinθ (2) When a inclined plane given a horizontal acceleration ‘b’ Since the body lies in an accelerating frame, an inertial force
(mb) acts on it in the opposite direction. Normal reaction R = mg cosθ + mb sinθ and ma = mg sin θ – mb cos θ ∴ a = g sinθ – b cosθ
3.16 Motion of Blocks in Contact.
Condition Free body diagram Equation Force and acceleration
F – f = m1a 1 2
Fa
m m=
+
f = m2a
2
1 2
m Ff
m m=
+
f = m1a 1 2
Fa
m m=
+
F – f = m2a 1
1 2
m Ff
m m=
+
F – f1 = m1a 1 2 3
Fa
m m m=
+ +
f1 – f2 = m2a 2 3
11 2 3
( )m m Ff
m m m
+=
+ +
f2 = m3a 3
21 2 3
m Ff
m m m=
+ +
m1
F f
m1a
m2 f
m2a
m1
f
m1a
m2 f
m2a
F
m1
f1
m1a
F
m2a
f1 m2
f2
m3 f2
m3a m1
m3 F
B
m2
C
m1
m2 F
A
B
m1
m2 F
A
B
mg cosθ
R
θ mg
a F
mg cosθ +mb sinθ
R
θ mg
a θ
θ
b
96 / Academic Excellence in Physics
f1 = m1a 1 2 3
Fa
m m m=
+ +
f2 – f1 = m2a 1
11 2 3
m Ff
m m m=
+ +
F – f2 = m3a 1 2
21 2 3
( )m m Ff
m m m
+=
+ +
m1
f1
m1a
m2a
f1 m2
f2
m3 f2
m3a
F
m1
m3 F
A B
m2
C
Newton’s Laws of Motion and Friction / 97
3.17 Motion of Blocks Connected by Mass Less String.
Condition Free body diagram Equation Tension and acceleration
1T m a= 1 2
Fa
m m=
+
2F T m a− = 1
1 2
m FT
m m=
+
1F T m a− = 1 2
Fa
m m=
+
2T m a= 2
1 2
m FT
m m=
+
1 1T m a= 1 2 3
Fa
m m m=
+ +
2 1 2T T m a− = 11
1 2 3
m FT
m m m=
+ +
2 3F T m a− = 1 22
1 2 3
( )m m FT
m m m
+=
+ +
1 1F T m a− = 1 2 3
Fa
m m m=
+ +
1 2 2T T m a− = 2 31
1 2 3
( )m m FT
m m m
+=
+ +
2 3T m a= 32
1 2 3
m FT
m m m=
+ +
m1
T
m1a
m2 F
m2a
T
m1
T
m1a
F
m1
T1
m1a
m3 F
m3a
T2
T2
m2a
T1m2
m1
T1
m1a
F
T2
m2a
T1m2
m3
m3a
T2
m2
m2a
T
m1
m2 F T
B
A
m1
m2 F T
B
A
m1
m3 F
A B
m2
C
T1 T2
m3 F
A B
m2
C
T1 T2 m1
98 / Academic Excellence in Physics
3.18 Motion of Connected Block Over a Pulley.
Condition Free body diagram Equation Tension and acceleration
1 1 1m a T m g= − 1 21
1 2
2m mT g
m m=
+
2 2 1m a m g T= − 1 22
1 2
4m mT g
m m=
+
2 12T T= 2 1
1 2
m ma g
m m
−= +
1 1 1m a T m g= − 1 2 31
1 2 3
2 [ ]m m mT g
m m m
+=
+ +
2 2 2 1m a m g T T= + −
1 32
1 2 3
2m mT g
m m m=
+ +
3 3 2m a m g T= − 1 2 33
1 2 3
4 [ ]m m mT g
m m m
+=
+ +
3 12T T= 2 3 1
1 2 3
[( ) ]m m m ga
m m m
+ −=
+ +
m1 m1a
T1
m1g
m2 m2a
T1
m2g
m1 m1a
T1
m1g
m3 m3a
T2
m3g
T1 T1
T3
P
T1
T1
T2
m1
m2 A
B
a
a
p
T3
m1
A
a
T1
T1
C
T2 B
m3
m2
m2 m2a
T1
m2g + T2
T1 T1
T2
Newton’s Laws of Motion and Friction / 99
Condition Free body diagram Equation Tension and
acceleration
When pulley have a finite mass M and radius R then tension in two segments of string are different
1 1 1m a m g T= − 1 2
1 2 2
m ma
Mm m
−=
+ +
2 2 2m a T m g= −
1 2
1
1 2
22
2
Mm m
T gM
m m
+ =+ +
Torque
1 2( )T T R I= − = α
1 2( )a
T T R IR
− =
21 2
1( )
2a
T T R MRR
− =
1 2 2Ma
T T− =
2 1
2
1 2
22
2
Mm m
T gM
m m
+ =+ +
1T m a= 2
1 2
ma g
m m=
+
2 2m a m g T= − 1 2
1 2
m mT g
m m=
+
1 1 sinm a T m g= − θ 2 1
1 2
sinm ma g
m m
− θ= +
2 2m a m g T= − 1 2
1 2
(1 sin )m mT g
m m
+ θ=
+
1 1sinT m g m a− α = 2 1
1 2
( sin sin )m ma g
m m
β − α=
+
2 2 sinm a m g T= β − 1 2
1 2
(sin sin )m mT g
m m
α + β=
+
m1 m1a
T1
m1g
m2 m2a
T2
m2g
T1 T2
α R
m1
A
T
P
m2 a
T
B
m1
m1a
T
m1
m1a T
θ
m1g sinθ
m2
m2g
T
m2a
m2
m2g
T
m2a
m2a T
m2g sinβ
m2
β
m1
A T
P
m2 a
T
θ B
m1
m1a T
α
m1g sinα a
m m
T T a
α β A B
M
T2
m2
B
T2
T1
A
m1
R
100 / Academic Excellence in Physics
Condition Free body
diagram
Equation Tension and
acceleration
1 1sinm g T m aθ − = 1
1 2
sinm ga
m m
θ=
+
2T m a= 1 2
1 2
24
m mT g
m m=
+
As 2
22
( )d x
dt
21
2
1 ( )2
d x
dt=
12 2
aa∴ =
a1 = acceleration of block A a2 = acceleration of block B
1T m a= 21
1 2
24
m ga a
m m= =
+
22
1 24m g
am m
=+
1 2
1 2
24
m m gT
m m=
+
2 2 22a
m m g T= −
1 1 1m a m g T= − 1 2
1 2
( )[ ]
m ma g
m m M
−=
+ +
2 2 2m a T m g= − 1 21
1 2
(2 )[ ]m m M
T gm m M
+=
+ +
1 2T T Ma− = 2 22
1 2
(2 )[ ]m m M
T gm m M
+=
+ +
3.19 Spring Balance and Physical Balance.
(1) Spring balance: When its upper end is fixed with rigid support and body of mass m hung from its lower end.
m1
A
P
m2
a T
θ
B
m1
m1a T
θ
m1g sinθ
m1
T1
m1g
m1a
m1 T
m1a
m2 T
m2a
m2
T2
m2g
m2a
M T2 T1
Ma
P
B
T T
a2
a1
m1
A
m2
M
m1
A
T1
m2
B
T1
a a
T2
T2 C
X Y
A B O
a b
m1
T1
m1
m1a
Newton’s Laws of Motion and Friction / 101
Spring is stretched and the weight of the body can be measured by the reading of spring balance R = W = mg.
(2) Physical balance: In physical balance actually we compare the mass of body in both the pans. Here we does not calculate the absolute weight of the body.
Here X and Y are the mass of the empty pan. (i) Perfect physical balance:
Weight of the pan should be equal i.e. X = Y and the needle must in middle of the beam i.e. a = b. Effect of frame of reference: If the physical balance is perfect then there will be no effect of frame of reference (either inertial or non-inertial)
(ii) False balance: When the masses of the pan are not equal then balance shows the error in measurement. False balance may be of two types: (a) If the beam of physical balance is horizontal (when the pans are empty) but the
arms are not equal X > Y and a < b
In this physical balance if a body of weight W is placed in pan X then to balance it we have to put a weight W1 in pan Y.
Now if the pans are changed then to balance the body we have to put a weight 2W in pan X.
True weight 1 2W W W=
(b) If the beam of physical balance is not horizontal (when the pans are empty) and the arms are equal i.e. X > Y and a = b
True weight 1 2
2W W
W+
=
3.20 Friction :Introduction.
If we slide or try to slide a body over a surface the motion is resisted by a bonding between the body and the surface. This resistance is represented by a single force and is called friction.
The force of friction is parallel to the surface and opposite to the direction of intended motion.
3.21 Types of Friction
(1) Static friction: The opposing force that comes into play when objects are at rest. (i) In this case static friction F = P.
(ii) Static friction is a self-adjusting force because it changes itself in accordance with the applied force.
(2) Limiting friction: The maximum value of static friction upto which body does not move is called limiting friction. (i) The magnitude of limiting friction between any two
bodies in contact is directly proportional to the normal reaction between them.
X
Y
A B
O
a b
P
R
F
mg
102 / Academic Excellence in Physics
F1 ∝ R or F1 = usR (ii) Direction of the force of limiting friction is always opposite to the direction in which one
body is at the verge of moving (iii) Coefficient of static friction:
(a) µs is called coefficient of static friction. (b) Dimension: [M0L0T0] (c) Unit: It has no unit. (d) Value of µs lies in between 0 and 1 (e) Value of µ depends on material and nature of surfaces in contact. (f) Value of µ does not depend upon apparent area of contact.
(3) Kinetic or dynamic friction: If the applied force sets the body in motion, the friction opposing the motion is called kinetic friction. (i) Kinetic friction depends upon the normal reaction.
Fk ∝ R or Fk = µkR where µk is called the coefficient of kinetic friction (ii) Kinetic friction is always lesser than limiting friction Fk < Fl ∴ µk < µs
Thus we require more force to start a motion than to maintain it against friction. This is because when motion has actually started, irregularities of one surface have little time to get locked again into the irregularities of the other surface.
(iv) Types of kinetic friction: (a) Sliding friction (b) Rolling friction � Rolling friction is directly proportional to the normal reaction (R) and inversely
proportional to the radius (r) of the rolling cylinder or wheel.
rolling r
RF
r= µ
µr is called coefficient of rolling friction. It would have the dimensions of length and would be measured in metre.
� Rolling friction is often quite small as compared to the sliding friction. � In rolling the surfaces at contact do not rub each other. � The velocity of point of contact with respect to the surface remains zero all the times.
3.22 Graph Between Applied Force and Force of Friction
(1) Part OA = static friction (Fs). (2) At point A = limiting friction (Fl). (3) Beyond A, the force of friction is seen to decrease
slightly. The portion BC = kinetic friction (Fk). (4) As the portion BC of the curve is parallel to x-axis
therefore kinetic friction does not change with the applied force.
3.23 Angle of Friction.
Angle of friction may be defined as the angle which the resultant of limiting friction and normal reaction makes with the normal reaction.
By definition angle θ is called the angle of friction tan θ = F
R
P
R
F
mg
θ S
A C B
Fl Fk
Fo
rce o
f fr
ictio
n
Applied force O
Fs
Newton’s Laws of Motion and Friction / 103
∴ tan θ = µ [As we know F
R= µ ]
or θ = tan–1(µ)
3.24 Angle of Repose
Angle of repose is defined as the angle of the inclined plane with horizontal such that a body placed on it is just begins to slide.
If α is called the angle of repose. α = θ i.e. angle of repose = angle of friction.
104 / Academic Excellence in Physics
4. Work, Energy and Power
4.1 Introduction
Work is said to be done when a force applied on the body displaces the body through a certain distance in the direction of force.
4.2 Work Done by a Constant Force
Let a constant force F���
be applied on the body such that it makes an angle θ with the horizontal and body is displaced through a distance s
Then work done by the force in displacing the body through a distance s is given by θθ cos)cos( FssFW == ( cos ) cosW F s Fs= θ = θ
or .W F s=��� ��
4.3 Nature of Work Done
Positive work Negative work
Positive work means that force (or its component) is parallel to displacement 0 90o o≤ θ <
The positive work signifies that the external force favours the motion of the body.
Negative work means that force (or its component) is opposite to displacement i.e.
90 180o o< θ ≤
The negative work signifies that the external force opposes the motion of the body.
4.4 Work Done by a Variable Force
When the magnitude and direction of a force varies with position, the work done by such a force for an infinitesimal displacement is given by .dW F ds=
��� ��
The total work done in going from A to B is . ( cos )B B
A AW F ds F ds= = θ∫ ∫
��� ��
Area under force displacement curve with proper algebraic sign represents work done by the force.
4.5 Dimension and Units of Work.
Dimension: As work = Force × displacement ∴ [W] = [Force] × [Displacement] = [MLT–2] × [L] = [ML2T–2] Units: The units of work are of two types 1 Joule = 1 Newton × 1 metre (SI unit) Erg [C.G.S.] 1Erg
= 1 Dyne × 1 cm.
4.6 Work Depends on Frame of Reference.
With change of frame of reference (inertial) force does not change while displacement may change. So the work done by a force will be different in different frames.
Direction of motion
θ
F
sθ
F
s
Direction of motion
Work, Energy and Power/ 105
Examples: If a person is pushing a box inside a moving train, the work done in the frame of train will .F s
��� ��
while in the frame of earth will be 0.( )F s s+��� �� ��
where 0s��
is the displacement of the train relative to the ground.
4.7 Energy
The energy of a body is defined as its capacity for doing work. (1) It is a scalar quantity. (2) Dimension: [ML2T–2] it is same as that of work or torque. (3) Units: Joule [S.I.], erg [C.G.S.] Practical units: electron volt (eV), Kilowatt hour (KWh), Calories (Cal) Relation between different units: 1 Joule = 107 erg 1 eV = 1.6 × 10–19 Joule 1 KWh = 3.6 × 106 Joule 1 Calorie = 4.18 Joule (4) Mass energy equivalence: The relation between the mass of a particle m and its equivalent
energy is given as 2mcE = where c = velocity of light in vacuum.
4.8 Kinetic Energy
The energy possessed by a body by virtue of its motion is called kinetic energy.
Let m = mass of the body, v = velocity of the body 212
W mv=
(1) Kinetic energy depends on frame of reference: The kinetic energy of a person of mass m,
sitting in a train moving with speed v, is zero in the frame of train but 212
mv in the frame of
the earth. (2) Work-energy theorem: It states that work done by a force acting on a body is equal to the
change produced in the kinetic energy of the body. This theorem is valid for a system in presence of all types of forces (external or internal,
conservative or non-conservative). (3) Relation of kinetic energy with linear momentum: As we know Momentum
22
EP mE
v= = .
(4) Various graphs of kinetic energy
E ∝ v2 m = constant
E ∝ P2
m = constant
E
v v
E
106 / Academic Excellence in Physics
1
Em
∝
P = constant
P E∝
m = constant
4.9 Stopping of Vehicle by Retarding Force.
(1) Comparison of stopping distance and time for two vehicles: Two vehicles of masses m1
and m2 are moving with velocities v1 and v2 respectively. When they are stopped by the same retarding force (F).
The ratio of their stopping distances 2
1 1 1 12
2 2 2 2
x E m v
x E m v= = and the ratio of their stopping time
1 1 1 1
2 2 2 2
t P m v
t P m v= =
If vehicles possess same velocities
v1 = v2 1 1
2 2
x m
x m= 1 1
2 2
t m
t m=
If vehicle possess same kinetic momentum
P1 = P2 2
1 1 1 2 22
2 2 1 2 1
22
x E P m m
x E m P m
= = =
1 1
2 2
1t P
t P= =
If vehicle possess same kinetic energy
E1 = E2 1 1
2 2
1x E
x E= = 1 11 1 1
2 2 22 2
2
2
m Et P m
t P mm E= = =
4.10 Potential Energy
Potential energy is defined only for conservative forces. In the space occupied by conservative forces every point is associated with certain energy which is called the energy of position or potential energy. Potential energy generally are of three types: Elastic potential energy, Electric potential energy and Gravitational potential energy etc.
(1) Change in potential energy: Change in potential energy between any two points is defined in the terms of the work done by the force in displacing the particle between these two points without any change in kinetic energy.
2
12 1 .
r
rU U F dr W− = − = −∫
� � ……(i)
(2) Potential energy curve: A graph plotted between the potential energy of a particle and its displacement from the centre of force is called potential energy curve. Negative gradient of the potential energy gives force.
∴ dU
Fdx
− =
E
m
P
E
Work, Energy and Power/ 107
E
En
erg
y
x = +a x = 0 x =– a
U
K
Position
O A B
(5) Types of equilibrium: If net force acting on a particle is zero, it is said to be in equilibrium.
For equilibrium 0dU
dx= , but the equilibrium of particle can be of three types:
Stable Unstable Neutral
When a particle is displaced slightly from a position, then a force acting on it brings it back to the initial position, it is said to be in stable equilibrium position.
When a particle is displaced slightly from a position, then a force acting on it tries to displace the particle further away from the equilibrium position, it is said to be in unstable equilibrium.
When a particle is slightly displaced from a position then it does not experience any force acting on it and continues to be in equilibrium in the displaced position, it is said to be in neutral equilibrium.
Potential energy is minimum. Potential energy is maximum. Potential energy is constant.
0dU
Fdx
= − = 0dU
Fdx
= − = 0dU
Fdx
= − =
2
2 positived U
dx=
i.e. rate of change of dU
dx is
positive.
2
2 negatived U
dx=
i.e. rate of change of dU
dx is
negative.
2
2 0d U
dx=
i.e. rate of change of dU
dx is
zero. Example: A marble placed at the bottom of a hemispherical bowl.
Example: A marble balanced on top of a hemispherical bowl.
Example: A marble placed on horizontal table.
4.11 Elastic Potential Energy.
(1) Restoring force and spring constant: When a spring is stretched or compressed from its normal position (x = 0) by a small distance x, then a restoring force is produced in the spring to bring it to the normal position.
According to Hooke’s law this restoring force is proportional to the displacement x and its direction is always opposite to the displacement.
i.e. F x∝ −��� ��
or F k x= −��� ��
…..(i)
where k is called spring constant. (2) Expression for elastic potential energy:
∴ Elastic potential energy 2
21 12 2 2
FU kx Fx
k= = =
Note:
� If spring is stretched from initial position x1 to final position x2 then work done =
Increment in elastic potential energy 2 22 1
1( )
2k x x= −
108 / Academic Excellence in Physics
(3) Energy graph for a spring: It mean kinetic energy changes parabolically w.r.t. position but total energy remain always constant irrespective to position of the mass.
4.12 Work Done in Pulling the Chain Against Gravity.
A chain of length L and mass M is held on a frictionless table with (1/n)th of its length hanging over the edge.
22MgL
Wn
=
4.13 Velocity of Chain While Leaving the Table
∴ Velocity of chain 2
11v gL
n
= −
4.14 Law of Conservation of Energy.
(1) Law of conservation of energy: For an isolated system or body in presence of conservative forces the sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depends upon time. This is known as the law of conservation of mechanical energy.
(2) Law of conservation of total energy: If the forces are conservative and non-conservative both, it is not the mechanical energy alone which is conserved, but it is the total energy, may be heat, light, sound or mechanical etc., which is conserved.
4.15 Power
Power of a body is defined as the rate at which the body can do the work.
Average power av.( )W W
Pt t
∆= =
∆. Instantaneous power inst.
.( )
dW F dsP
dt dt= =
� �
[As .dW F ds=� �
]
inst .P F v=� �
. i.e. power is equal to the scalar product of force with velocity. (1) Dimension: [P] = [ML2T–3] (2) Units: Watt or Joule/sec [S.I.] Practical units: Kilowatt (kW), Mega watt (MW) and Horse power (hp) Relations between different units: 71 1 / sec 10 / secwatt Joule erg= = 1 746hp Watt=
(3) The slope of work time curve gives the instantaneous power. As P = dW/dt = tanθ
(4) Area under power time curve gives the work done as dW
Pdt
=
∴ W P dt= ∫
∴ W = Area under P-t curve
Work, Energy and Power/ 109
4.16 Collision
Collision is an isolated event in which a strong force acts between two or more bodies for a short time as a result of which the energy and momentum of the interacting particle change.
In collision particles may or may not come in real touch (3) Types of collision: (i) On the basis of conservation of kinetic energy.
Perfectly elastic collision Inelastic collision Perfectly inelastic collision
If in a collision, kinetic energy after collision is equal to kinetic energy before collision, the collision is said to be perfectly elastic.
If in a collision kinetic energy after collision is not equal to kinetic energy before collision, the collision is said to inelastic.
If in a collision two bodies stick together or move with same velocity after the collision, the collision is said to be perfectly inelastic.
Coefficient of restitution e = 1 Coefficient of restitution 0 < e < 1 Coefficient of restitution e = 0
(KE)final = (KE)initial
Here kinetic energy appears in other forms. In some cases (KE)final < (KE)initial such as when initial KE is converted into internal energy of the product (as heat, elastic or excitation) while in other cases (KE)final > (KE)initial such as when internal energy stored in the colliding particles is released.
The term 'perfectly inelastic' does not necessarily mean that all the initial kinetic energy is lost, it implies that the loss in kinetic energy is as large as it can be. (Consistent with momentum conservation).
Examples: (1) Collision between atomic particles (2) Bouncing of ball with same velocity after the collision with earth.
Examples: (1) Collision between two billiard balls. (2) Collision between two automobile on a road. In fact all majority of collision belong to this category.
Example: Collision between a bullet and a block of wood into which it is fired. When the bullet remains embeded in the block.
4.17 Perfectly Elastic Head on Collision.
Let two bodies of masses m1 and m2 moving with initial velocities u1 and u2 in the same direction and they collide such that after collision their final velocities are v1 and v2 respectively.
According to law of conservation of momentum and conservation of kinetic energy.
Note:
� The ratio of relative velocity of separation and relative velocity of approach is defined as
coefficient of restitution. 2 1
1 2
v ve
u u
−=
− or 2 1 1 2( )v v e u u− = −
� For perfectly elastic collision e = 1 ∴ 2 1 1 2v v u u− = − (As shown in eq. (vi)
� For perfectly inelastic collision e = 0 ∴ 2 1 0v v− = or 2 1v v= It means that two body stick together and move with same velocity. � For inelastic collision 0 < e < 1 ∴ 2 1 1 2( )v v e u u− = − In short we can say that e is the degree of elasticity of collision and it is dimension less
quantity.
Before collision After collision
m1
u1 u2 m2 m1
v1 v2 m2
110 / Academic Excellence in Physics
1 2 2 21 1
1 2 1 2
2m m m uv u
m m m m
−= +
+ + ……(vii)
2 1 1 12 2
1 2 1 2
2m m m uv u
m m m m
−= +
+ + ……(viii)
� When two bodies of equal masses undergo head on elastic collision, their velocities get interchanged.
(2) Kinetic energy transfer during head on elastic collision. Fractional decrease in kinetic energy
1 22
1 2 1 2
4( ) 4
K m m
K m m m m
∆=
− + …..(iv)
Note:
� Greater the difference in masses less will be transfer of kinetic energy and vice versa � Transfer of kinetic energy in head on elastic collision (when target is at rest) is maximum
when the masses of particles are equal.
4.18 Perfectly Elastic Oblique Collision.
Collision is said to be elastic oblique if after collision directions of Bodies are not along a straight line.
4.19 Head on Inelastic Collision.
(1) Velocity after collision: Let two bodies A and B collide inelastically and coefficient of restitution is e.
1 2 21 1 2
1 2 1 2
(1 )m em e mv u u
m m m m
− += +
+ +
Similarly 1 2 12 1 2
1 2 1 2
(1 )e m m e mv u u
m m m m
+ −= + + +
(2) Ratio of velocities after inelastic collision: A sphere of mass m moving with velocity u hits in elastically with another stationary sphere of same mass.
1
2
11
v e
v e
−=
+
(3) Loss in kinetic energy: Loss (∆K) = 2 21 21 2
1 2
1(1 )( )
2m m
e u um m
− −
+
4.20 Rebounding of Ball After Collision With Ground.
If a ball is dropped from a height h on a horizontal floor, then it strikes with the floor with a speed.
0 02v gh=
and it rebounds from the floor with a speed
1 0v ev= 02e gh= v0 v1 v2
h0
h1 h2
t0 t1 t2
Work, Energy and Power/ 111
velocity after collision
As velocity before collision
e
=
(1) Height of the ball after nth rebound: The height after nth rebound will be
∴ 02 heh n
n =
(3) Total distance travelled by the ball before it stops bouncing: 2
0 2
11
eH h
e
+= −
(4) Total time taken by the ball to stop bouncing
∴ 01 21
e hT
e g
+ = −
112 / Academic Excellence in Physics
5. Rotational Motion
5.1 Introduction
Rigid body: A rigid body is a body that can rotate with all the parts locked together and without any change in its shape.
5.2 Centre of Mass.
Centre of mass of a system is a point that moves as though all the mass were concentrated there and all external forces were applied there.
(1) Position vector of centre of mass for n particle system: If a system consists of n
particles of masses 1 2 3, , ...... nm m m m , whose positions vectors are 1 2 3, , ........ nr r r r→ → → →
respectively then position vector of centre of mass
1 1 2 2 3 3
1 2 3
........................
n n
n
m r m r m r m rr
m m m m
→ → → →→ + + +
=+ + +
If two masses are equal i.e. 1 2m m= , then position vector of centre of mass 1 2
2r r
r
→ →→ +
=
(2) Important points about centre of mass (i) The position of centre of mass is independent of the co-ordinate system chosen. (ii) The position of centre of mass depends upon the shape of the body and distribution
of mass. (iii) In symmetrical bodies in which the distribution of mass is homogenous, the centre of
mass coincides with the geometrical centre or centre of symmetry of the body.
Centre of mass of Cone or pyramid lise on the axis of the cone at point distance 34h
from the vertex where h is the height of cone (iv) The centre of mass changes its position only under the translatory motion. There is
no effect of rotatory motion on centre of mass of the body. (v) If the origin is at the centre of mass, then the sum of the moments of the masses of
the system about the centre of mass is zero i.e. 0i im r→
Σ = .
(vi) If a system of particles of masses 1 2 3, , ,......m m m move with velocities 1 2 3, , ,......v v v
then the velocity of centre of mass i icm
i
m vv
m
∑=
∑.
(vii) If a system of particles of masses 1 2 3, , ,......m m m move with accelerations
1 2 3, , ,......a a a then the acceleration of centre of mass i icm
i
m aA
m
∑=
∑
Rotational Motion / 113
(viii) If r→
is a position vector of centre of mass of a system then velocity of centre of mass →
→
=cm
d rv
dt
(ix) Acceleration of centre of mass 2
2
→ →→
= =cm
cm
d v d rA
dt dt
(x) Force on a rigid body 2
2
→→ →
= =cm
d rF M A M
dt
(xi) For an isolated system external force on the body is zero 0→ → = =
cm
dF M v
dt
⇒ constant→
=cmv . i.e., centre of mass of an isolated system moves with uniform velocity along a straight-line path.
5.3 Angular Displacement.
It is the angle described by the position vector ��
r about the axis of rotation.
Angular displacement Linear displacement ( )( )
Radius ( )s
rθ =
(1) Unit : radian (2) Dimension : [M0L0T0]
(3) Vector form S r→ → →
= θ×
i.e., angular displacement is a vector quantity whose direction is given by right hand rule. It is also known as axial vector. For anti-clockwise sense of rotation direction of θ is perpendicular to the plane, outward and along the axis of rotation and vice-versa.
(4) 2 radian 360 1 revolution.π = ° =
5.4 Angular Velocity.
The angular displacement per unit time is defined as angular velocity.
If a particle moves from P to Q in time ∆t, t
∆θω =
∆ where ∆θ is the angular displacement.
(1) Instantaneous angular velocity 0
limt
d
t dt∆ →
∆θ θω = =
∆
(2) Unit : Radian/sec (3) Dimension : [M0L0T–1] which is same as that of frequency.
(4) Vector form v r→ → →
= ω × [where v→
= linear velocity, r→
= radius vector] →
ω is a axial vector, whose direction is normal to the rotational plane and its direction is given by right hand screw rule.
(5) 2
2 nT
πω = = π [where T = time period, n = frequency]
114 / Academic Excellence in Physics
5.5 Angular Acceleration.
The rate of change of angular velocity is defined as angular acceleration. If particle has angular velocity ω1 at time t1 and angular velocity ω2 at time t2 then,
Angular acceleration 2 1
2 1t t
→ →→ ω − ωα =
−
(1) Instantaneous angular acceleration 2
20lim
t
d d
t dt dt
→ →→
∆ →
∆ω ω θα = = =
∆.
(2) Unit : rad/sec2 (3) Dimension : [M0L0T–2]. (4) If α = 0, circular or rotational motion is said to be uniform.
(5) Relation between angular acceleration and linear acceleration a r→ → →
= α× .
(6) It is an axial vector whose direction is along the change in direction of angular velocity i.e. normal to the rotational plane, outward or inward along the axis of rotation (depends upon the sense of rotation).
5.6 Equations of Linear Motion and Rotational Motion.
Rotational Motion
If angular acceleration is 0, ω = constant and θ = ωt
If angular acceleration α = constant then
(i) 1 2( )2
tω + ω
θ =
(ii) 2 1
t
ω − ωα =
(iii) 2 1 tω = ω + α
(iv) 21
12
t tθ = ω + α
(v) 2 22 1 2ω = ω + αθ
(vi) 1 (2 1)2nth nα
θ = ω + −
If acceleration is not constant, the above equation will not be applicable. In this case
(i) d
dt
θω =
Rotational Motion / 115
(ii) 2
2
d d
dt dt
ω θα = =
(iii) d dω ω = α θ
5.7 Moment of Inertia.
Moment of inertia plays the same role in rotational motion as mass plays in linear motion. It is the property of a body due to which it opposes any change in its state of rest or of uniform rotation.
(1) Moment of inertia of a particle I = mr2; where r is the perpendicular distance of particle from rotational axis.
(2) Moment of inertia of a body made up of number of particles (discrete distribution) 2 2 2
1 1 2 2 3 3 .......I m r m r m r= + + +
(3) Moment of inertia of a continuous distribution of mass, dI = dmr2 i.e., 2I r dm= ∫
(4) Dimension : [ML2T0] (5) S.I. unit : kgm2. (6) Moment of inertia depends on mass, distribution of mass and on the position of axis of
rotation. (7) Moment of inertia is a tensor quantity.
5.8 Radius of Gyration.
Radius of gyration of a body about a given axis is the perpendicular distance of a point from the axis, where if whole mass of the body were concentrated, the body shall have the same moment of inertia as it has with the actual distribution of mass.
When square of radius of gyration is multiplied with the mass of the body gives the moment of inertia of the body about the given axis.
I = Mk2 or I
kM
= .
Here k is called radius of gyration.
∴ 2 2 2 2
1 2 3 ........... nr r r rk
n
+ + + +=
Note:
� For a given body inertia is constant whereas moment of inertia is variable.
5.9 Theorem of Parallel Axes.
Moment of inertia of a body about a given axis I is equal to the sum of moment of inertia of the body about an axis parallel to given axis and passing through centre of mass of the body Ig and
a
I
G
IG
116 / Academic Excellence in Physics
X
Z
Y
Ma2 where M is the mass of the body and a is the perpendicular distance between the two axes. I = Ig + Ma2
5.10 Theorem of Perpendicular Axes.
According to this theorem the sum of moment of inertia of a plane lamina about two mutually perpendicular axes lying in its plane is equal to its moment of inertia about an axis perpendicular to the plane of lamina and passing through the point of intersection of first two axes.
Iz = Ix + Iy Note:
� In case of symmetrical two-dimensional bodies as moment of inertia for all axes passing through the centre of mass and in the plane of body will be same so the two axes in the plane of body need not be perpendicular to each other.
5.12 Analogy between Tranlatory Motion and Rotational Motion.
Translatory motion Rotatory motion
Mass (m) Moment of Inertia
(I)
Linear momentum
P = mv
2P mE=
Angular Momentum
L = Iω
2L IE=
Force F = ma Torque τ = Iα
Kinetic energy 212
E mv=
2
2P
Em
=
212
E I= ω
2
2L
EI
=
Rotational Motion / 117
5.13 Moment of Inertia of Some Standard Bodies about Different Axes
Body Axis of
Rotation Figure
Moment of
inertia k
k2/R2
Ring
(Cylindrical shell)
About an axis passing through C.G. and perpendicular to its plane
MR2
R
1
Ring About its diameter
212
MR 2
R 12
Ring About a tangential axis in its own plane
232
MR 32
R 32
Ring About a tangential axis perpendicular to its own plane
22MR 2R 2
Disc
(Solid cylinder)
About an axis passing through C.G. and perpendicular to its plane
212
MR 2
R 12
118 / Academic Excellence in Physics
Body Axis of
Rotation Figure
Moment of
inertia k
k2/R2
Disc About its Diameter
214
MR 2R 1
4
Disc
About a tangential axis in its own plane
254
MR 52
R 54
Disc
About a tangential axis perpendicular to its own plane
232
MR 32
R 32
Annular disc inner radius = R1 and outer radius = R2
Passing through the centre and perpendicular to the plane
2 21 2[ ]
2M
R R+ – –
Solid cylinder
About an axis passing through its C.G. and perpendicular to its own axis
2 2
12 4L R
M
+
2 2
12 4L R
+
Solid cylinder
About the diameter of one of faces of the cylinder
2 2
3 4L R
M
+
2 2
3 4L R
+
R2
R1
Rotational Motion / 119
Body Axis of
Rotation Figure
Moment of
inertia k
k2/R2
Solid Sphere About its diametric axis
22
5MR 2
5R
25
Solid sphere
About a tangential axis
275
MR 75
R 75
Spherical shell
About its diametric axis
223
MR 23
R 23
Spherical shell About a tangential
axis
253
MR 53
R 53
Long thin rod
About on axis passing through its centre of mass and perpendicular to the rod.
2
12ML
12L
Long thin rod
About an axis passing through its edge and perpendicular to the rod
2
3ML
3L
5.14 Torque.
120 / Academic Excellence in Physics
If the particle rotating in xy plane about the origin under the effect of force F→
and at any instant
the position vector of the particle is r→
then,
Torque →
τ = r F→ →
× τ = r F sin φ
[where φ is the angle between the direction of r→
and F→
] (1) Torque is an axial vector. i.e., its direction is always
perpendicular to the plane containing vector r→
and F→
in accordance with right hand screw rule. For a given figure the sense of rotation is anti-clockwise so the direction of torque is perpendicular to the plane, outward through the axis of rotation.
i.e. Torque = Force × Perpendicular distance of line of action of force from the axis of rotation.
Torque is also called as moment of force and d is called moment or lever arm.
(2) Unit : Newton-metre (M.K.S.) and Dyne-cm (C.G.S.) (3) Dimension : [ML2T–2]. (4) A body is said to be in rotational equilibrium if resultant torque acting on it is zero i.e.
0→
Σ τ = . (5) Torque is the cause of rotatory motion and in rotational motion it plays same role as
force plays in translatory motion i.e., torque is rotational analogue of force. This all is evident from the following correspondences between rotatory and translatory motion.
Rotatory Motion Translatory Motion
I→ →
τ = α F m a→ →
=
W d→ →
= τ⋅ θ∫ W F ds→ →
= ⋅∫
P→ →
= τ⋅ ω P F v→ →
= ⋅
dL
dt
→→
τ = dPF
dt
→→
=
5.15 Couple
(1) A couple is defined as combination of two equal but oppositely directed force not acting along the same line. The effect of couple is known by its moment of couple
or torque by a couple τ r F→ → →
= × .
F
F
r
F�
F cos φ F sin φ
X
φ
φ
r�
P
d 90o
Y
Rotational Motion / 121
(2) Work done by torque in twisting the wire 212
W C= θ .
Where θτ C= ; C is known as twisting coefficient or couple per unit twist.
5.16 Translatory and Rotatory Equilibrium
Forces are equal and act along the same line.
ΣF = 0 and Στ = 0
Body will remain stationary if initially it was at rest.
Forces are equal and does not act along the same line.
ΣF = 0 and Στ ≠ 0
Rotation i.e. spinning.
Forces are unequal and act along the same line.
ΣF ≠ 0 and Στ = 0
Translation i.e. slipping or skidding.
Forces are unequal and does not act along the same line.
ΣF ≠ 0 and Στ ≠ 0
Rotation and translation both i.e. rolling.
5.17 Angular Momentum
The moment of linear momentum of a body with respect to any axis of rotation is known as
angular momentum. If P→
is the linear momentum of particle and r→
its position vector from the point of rotation then angular momentum.
L r P→ → →
= ×
ˆsinL r P n→
= φ
Angular momentum is an axial vector i.e. always directed perpendicular to the plane of rotation and along the axis of rotation.
(1) S.I. Unit : kg-m2-s–1 or J-sec. (2) Dimension : [ML2T–1] and it is similar to Planck’s constant (h). (3) Angular momentum = (Linear momentum) × (Perpendicular distance of line of action of
force from the axis of rotation)
(4) In vector form L I→ →
= ω
(5) From L I→ →
= ω ∴ d L d
Idt dt
→ →
ω= = I
→ →
α = τ
[Rotational analogue of Newton's second law]
F F
F
F
F2 F1
F2
F1
122 / Academic Excellence in Physics
(6) If a large torque acts on a particle for a small time then 'angular impulse' of torque is
given by 2
1
t
avt
J dt dt→ → →
= τ = τ∫ ∫
∴ Angular impulse = Change in angular momentum
5.18 Law of Conservation of Angular Momentum
If the net external torque on a particle (or system) is zero then 0d L
dt
→
=
i.e. 1 2 3 .......L L L L→ → → →
= + + + = constant.
Angular momentum of a system (may be particle or body) remains constant if resultant torque acting on it zero.
As L = Iω so if 0→
τ = then Iω = constant.
5.19 Work, Energy and Power for Rotating Body.
(1) Work : If the body is initially at rest and angular displacement is θd due to torque then work done on the body.
W d= τ θ∫
(2) Kinetic energy : The energy, which a body has by virtue of its rotational motion is called rotational kinetic energy.
Rotational kinetic energy Analogue to translatory kinetic
energy
212RK I= ω 21
2TK mv=
12RK L= ω 1
2TK Pv=
2
2R
LK
I=
2
2T
PK
m=
(3) Power: Rate of change of kinetic energy is defined as power
In vector form Power → →
= τ⋅ ω
5.20 Slipping, Spinning and Rolling.
(1) Slipping : When the body slides on a surface without rotation then its motion is called slipping motion.
In this condition friction between the body and surface F = 0.
Rotational Motion / 123
Body possess only translatory kinetic energy 212TK mv= .
(2) Spinning : When the body rotates in such a manner that its axis of rotation does not move then its motion is called spinning motion. In this condition axis of rotation of a body is fixed.
In spinning, body possess only rotatory kinetic energy 212RK I= ω .
(3) Rolling : If in case of rotational motion of a body about a fixed axis, the axis of rotation also moves, the motion is called combined translatory and rotatory. Example : (i) Motion of a wheel of cycle on a road. (ii) Motion of football rolling on a surface.
In this condition friction between the body and surface 0≠F .
Body possesses both translational and rotational kinetic energy.
Net kinetic energy = (Translatory + Rotatory) kinetic energy.
5.21 Rolling Without Slipping
In case of combined translatory and rotatory motion if the object rolls across a surface in such a way that there is no relative motion of object and surface at the point of contact, the motion is called rolling without slipping.
Friction is responsible for this type of motion but work done or dissipation of energy against friction is zero as there is no relative motion between body and surface at the point of contact.
Rolling motion of a body may be treated as a pure rotation about an axis through point of contact with same angular velocity ω. [v = Rω]
Linear velocity of different points in rolling : In case of rolling, all points of a rigid body have same angular speed but different linear speed.
Let A, B, C and D are four points then their velocities are shown in the following figure.
5.22 Rolling on an Inclined Plane.
Translation
v
v
v
v
A
B
C D
Rotation
v
v v
v = 0
A
B
C D
Rolling
2v
v = 0
B
C
D
√2 v
+
v
ω
=
124 / Academic Excellence in Physics
When a body of mass m and radius R rolls down on inclined plane of height ‘h’ and angle of inclination θ, it loses potential energy. However it acquires both linear and angular speeds and hence, gain kinetic energy of translation and that of rotation.
(1) Velocity at the lowest point : 2
2
2
1
ghv
k
R
=+
(2) Acceleration in motion : From equation 2 2 2v u aS= +
By substituting 2
2
20, and
sin 1
h ghu S v
k
R
= = =θ +
we get
2
2
sin
1
ga
k
R
θ=
+
(3) Time of descent : From equation v = u + at
By substituting u = 0 and value of v and a from above expressions 2
2
1 21
sinh k
tg R
= + θ
5.25 Motion of Connected Mass.
A point mass is tied to one end of a string which is wound round the solid body [cylinder, pulley, disc]. When the mass is released, it falls vertically downwards and the solid body rotates unwinding the string
m = mass of point-mass, M = mass of a rigid body R = radius of a rigid body, I = moment of inertia of rotating body
(1) Downwards acceleration of point mass
21
ga
I
mR
=+
(2) Tension in string 2
IT mg
I mR
= +
(3) Velocity of point mass 2
2
1
ghv
I
mR
=+
(4) Angular velocity of rigid body 2
2mgh
I mRω =
+
m
T h
h
B
θ
S
Translation
C
Rotation
Gravitation / 125
6. Gravitation
6.1 Newton's law of Gravitation
Newton's law of gravitation states that every body in this universe attracts every other body with a force, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The direction of the force is along the line joining the particles.
Thus the magnitude of the gravitational force F that two particles of masses m1 and m2
separated by a distance r exert on each other is given by 1 22
m mF
r∝ .
or 1 22
m mF G
r=
Also clear that 12F→
= – 21F→
. Which is Newton's third law of motion. Here G is constant of proportionality which is called 'Universal gravitational constant'. (i) The value of G is 6.67 × 10–11 N–m2 kg–2 in S.I. and 6.67 × 10–8 dyne- cm2-g–2 in C.G.S.
system. (ii) Dimensional formula [M–1L3T–2]. (iii) The value of G does not depend upon the nature and size of the bodies. (iv) It also does not depend upon the nature of the medium between the two bodies.
6.2 Properties of Gravitational Force
(1) It is always attractive in nature. (2) It is independent of the medium between the particles. (3) It is found true for interplanetary to inter atomic distances. (4) It is a central force i.e. acts along the line joining the centres of two interacting bodies. (5) The principle of superposition is valid. (6) It is the weakest force in nature . (7) It is a conservative force.
� The law of gravitation is stated for two point masses
6.3 Acceleration Due to Gravity.
The force of attraction exerted by the earth on a body is called gravitational pull or gravity. The acceleration produced in the motion of a body under the effect of gravity is called
acceleration due to gravity, it is denoted by g. If M = mass of the earth and R = radius of the earth and g is the acceleration due to gravity,
then
∴ 2
GMg
R=
43
GR= πρ
(i) Its value depends upon the mass radius and density of planet and it is independent of mass, shape and density of the body placed on the surface of the planet.
126 / Academic Excellence in Physics
(ii) Acceleration due to gravity is a vector quantity and its direction is always towards the centre of the planet.
(iii) Dimension [g] = [LT–2] (iv) It’s average value is taken to be 9.8 m/s2 or 981 cm/sec2 or 32 feet/sec2, on the surface of
the earth at mean sea level.
6.4 Variation in g Due to Shape of Earth.
Earth is elliptical in shape. The equatorial radius is about 21 km longer than polar radius.
At equator 2e
e
GMg
R=
At poles 2p
p
GMg
R=
∴ pole equatorg g>
Therefore the weight of body increases as it is taken from equator to the pole.
6.5 Variation in g with Height.
Acceleration due to gravity at height h from the surface of the earth
2'( )
GMg
R h=
+
Also 2
'R
g gR h
= +
2
2
Rg
r= [As r = R + h]
(i) If Rh << g′2
1h
gR
= −
(ii) If Rh << Percentage decrease 2100% 100%
g h
g R
∆× = ×
6.6 Variation in g With Depth
Acceleration due to gravity at depth d from the surface of the earth
4
( )3
g G R d′ = πρ −
also 1d
g gR
′ = −
(i) The value of g decreases on going below the surface of the earth. (ii) The acceleration due to gravity at the centre of earth becomes zero.
(iii) Percentage decrease 100% 100%g d
g R
∆× = ×
Gravitation / 127
(iv) The rate of decrease of gravity outside the earth (if h << R) is double to that of inside the earth.
6.7 Variation in g Due to Rotation of Earth
If the body of mass m lying at point P, whose latitude is λ, then due to rotation of earth its apparent acceleration can be given by g′ = g – ω2 R cos2 λ.
� The latitude at a point on the surface of the earth is defined as the angle, which the line joining that point to the centre of earth makes with equatorial plane. It is denoted by λ.
� For the poles λ = 90° and for equator λ = 0° (i) gpole = g (ii) gequator = g – ω2R (iii) If ω is the angular velocity of rotation of earth for which a body at the equator will
become weightless (g′ = 0)
g
Rω =
or time period of rotation of earth 2
2R
Tg
π= = π
ω
� If earth starts rotating 17 times faster then all objects on equator will become weightless.
6.8 Inertial and Gravitational Masses.
(1) Inertial mass: It is the mass of the material body, which measures its inertia.
i
Fm
a=
Hence inertial mass of a body may be measured as the ratio of the magnitude of the external force applied on it to the magnitude of acceleration produced in its motion. (i) Gravity has no effect on inertial mass of the body. (ii) It is proportional to the quantity of matter contained in the body. (iii) When a body moves with velocity v, its inertial mass is given by
0
2
21
mm
v
c
=
−
, where m0 = rest mass of body, c = velocity of light in vacuum,
(2) Gravitational Mass: Gravitational mass of a body may be measured as the ratio of the magnitude of the gravitational force applied on it to the magnitude of acceleration due to gravity.
Spring balance measure gravitational mass and inertial balance measure inertial mass.
6.9 Gravitational Field.
The space surrounding a material body in which gravitational force of attraction can be experienced is called its gravitational field.
128 / Academic Excellence in Physics
Gravitational field intensity: The intensity of the gravitational field of a material body at any point in its field is defined as the force experienced by a unit mass (test mass) placed at that
point. If a test mass m at a point in a gravitational field experiences a force F���
then F
Im
=
�����
.
(i) It is a vector quantity and is always directed towards the centre of gravity of body whose gravitational field is considered.
(ii) Units: Newton/kg or m/s2 (iii) Dimension: [M0LT–2]
(iv) 2
GMI
r=
(v) 1 2 3 ........netI I I I= + + +���� ��� ��� ���
6.10 Gravitational Field Intensity for Different Bodies
(1) Intensity due to uniform solid sphere
Outside the
surface
r > R
On the surface
r = R
Inside the
surface
r < R
2
GMI
r= 2
GMI
R= 3
GMrI
R=
(2) Intensity due to spherical shell
Outside the
surface
r > R
On the surface
r = R
Inside the
surface
r < R
2
GMI
r= 2
GMI
R= I = 0
(3) Intensity due to uniform circular ring
At a point on its axis At the centre of the ring
2 2 3 / 2( )GMr
Ia r
=+
I = 0
R
I
O r = R r
R
I
O r = R r
GM/R2
a P
r
I
Gravitation / 129
(4) Intensity due to uniform disc
At a point on its axis At the centre of the disc
2 2 2
2 1 1GMrI
a r r a
= −
+
or 2
2(1 cos )
GMI
a= − θ
I = 0
6.11 Gravitational Potential.
At a point in a gravitational field potential V is defined as negative of work done per unit mass in shifting a test mass from some reference point (usually at infinity) to the given point i.e.,
W
Vm
= −.F dr
m= −∫
��� ��
.I dr= −∫�� ��
∴ dV
Idr
= −
Negative sign indicates that the direction of intensity is in the direction where the potential decreases.
(i) It is a scalar quantity. (ii) Unit: Joule/kg or m2/sec2 (iii) Dimension: [M0L2T–2] (iv) If the field is produced by a point mass then
∴ Gravitational potential GM
Vr
= −
(v) Gravitational potential difference: It is defined as the work done to move a unit mass from one point to the other in the gravitational field. The gravitational potential difference in bringing unit test mass m from point A to point B under the gravitational influence of source mass M is
1 1A B
B A
B A
WV V V GM
m r r→
∆ = − = = − −
(vi) Potential due to large numbers of particle is given by scalar addition of all the potentials. 1 2 3 ..........V V V V= + + +
6.12 Gravitational Potential for Different Bodies
(1) Potential due to uniform ring At a point on its axis At the centre
2 2
GMV
a r= −
+ GM
Va
= −
a
P
r
I θ
a
P
r
130 / Academic Excellence in Physics
(2) Potential due to spherical shell
Outside the
surface
r > R
On the surface
r = R
Inside the surface
r < R
GMV
r
−= GM
VR
−= GM
VR
−=
(3) Potential due to uniform solid sphere Outside the
surface
r > R
On the surface
r = R
Inside the surface
r < R
GMV
r
−= surface
GMV
R
−=
2
32GM r
VR R
− = −
at the centre (r = 0) 3
2centre
GMV
R
−= (max.)
Vcentre = 32 surfaceV
6.13 Gravitational Potential Energy
The gravitational potential energy of a body at a point is defined as the amount of work done in bringing the body from infinity to that point against the gravitational force.
GMm
Wr
= −
This work done is stored inside the body as its gravitational potential energy
∴ GMm
Ur
= −
(i) Potential energy is a scalar quantity. (ii) Unit: Joule (iii) Dimension: [ML2T–2] (iv) Gravitational potential energy is always negative in the gravitational field because the
force is always attractive in nature. (v) If r = ∞ then it becomes zero (maximum) (vi) In case of discrete distribution of masses
Gravitational potential energy 1 2 2 3
12 23
........i
Gm m Gm mU u
r r
= ∑ = − + +
–
GM/R
R
V
O r = R r
–
3GM/2R
R
V
O r = R
r
Gravitation / 131
(vii) If the body of mass m is moved from a point at a distance r1 to r2(r1 > r2) then change in
potential energy 1 2
1 1U GMm
r r
∆ = −
(viii) Relation between gravitational potential energy and potential U = mV (ix) Gravitational potential energy of a body at height h from the earth surface is given by
2
1h
GMm gR m mgRU
hR h R hR
= − = − = −+ + +
6.14 Work Done Against Gravity.
If the body of mess m is moved from the surface of earth to a point at distance h above the surface of earth, then change in potential energy or work done against gravity will be
2 11
GMmh mghW
hhR
RR
= = ++
(i) When the distance h is not negligible and is comparable to radius of the earth, then we will use above formula.
(ii) If h = R then 12
W mgR=
(iv) If h is very small as compared to radius of the earth then term h/R can be neglected
1 /
mghW mgh
h R= =
+
6.15 Escape Velocity.
The minimum velocity with which a body must be projected up so as to enable it to just overcome the gravitational pull, is known as escape velocity.
If ve is the required escape velocity, then
2e
GMv
R= ⇒ 2ev gR=
(i) Escape velocity is independent of the mass and direction of projection of the body. (ii) For the earth ve = 11.2 km/sec (iii) A planet will have atmosphere if the velocity of molecule in its atmosphere is lesser than
escape velocity. This is why earth has atmosphere while moon has no atmosphere
6.16 Kepler’s Laws of Planetary Motion
(1) The law of Orbits: Every planet moves around the sun in an elliptical orbit with sun at one of the foci.
(2) The law of Area: The line joining the sun to the planet sweeps out equal areas in equal interval of time. i.e. areal velocity is constant. According to this law planet will move
132 / Academic Excellence in Physics
slowly when it is farthest from sun and more rapidly when it is nearest to sun. It is similar to law of conservation of angular momentum.
Areal velocity 2
dA L
dt m=
(3) The law of periods: The square of period of revolution (T) of any planet around sun is directly proportional to the cube of the semi-major axis of the orbit.
T2 ∝ a3 or 3
2 1 2
2r r
T+ ∝
where a = semi-major axis r1 = Shortest distance of planet from sun (perigee). r2 = Largest distance of planet from sun (apogee).
� Kepler's laws are valid for satellites also.
6.17 Velocity of a Planet in Terms of Eccentricity
Speeds of planet at apogee and perigee are
2 1
1a
GM ev
a e
− = + ,
2 11p
GM ev
a e
+ = −
� Angular momentum of a planet or satellite is always constant irrespective of shape of orbit.
6.18 Orbital Velocity of Satellite.
⇒ GMv
r= [r = R + h]
(i) Orbital velocity is independent of the mass of the orbiting body. (ii) Orbital velocity depends on the mass of planet and radius of orbit. (iii) Orbital velocity of the satellite when it revolves very close to the surface of the planet
GMv gR
R= = ≈ 8 km/sec
6.19 Time Period of Satellite
( )3
22R h
Tg R
+= π
3 / 2
2 1R h
g R
= π +
[As r = R + h]
(i) Time period is independent of the mass of orbiting body (ii) T2 ∝ r3 (Kepler’s third law)
(iii) Time period of nearby satellite, 2R
Tg
= θ
For earth T = 84.6 minute ≈ 1.4 hr.
B Sun
a
F
r1
r2
D
Apogee Perigee
A
E
C
Gravitation / 133
6.20 Height of Satellite
1 / 32 2
24T g R
h R
= − π
6.21 Geostationary Satellite.
The satellite which appears stationary relative to earth is called geostationary or geosynchronous satellite, communication satellite.
A geostationary satellite always stays over the same place above the earth. The orbit of a geostationary satellite is known as the parking orbit.
(i) It should revolve in an orbit concentric and coplanar with the equatorial plane. (ii) It sense of rotation should be same as that of earth. (iii) Its period of revolution around the earth should be same as that of earth. (iv) Height of geostationary satellite from the surface of earth h = 6R = 36000 km (v) Orbital velocity v = 3.08 km/sec (vi) Angular momentum of satellite depend on both the mass of orbiting and planet as well as
the radius of orbit.
6.22 Energy of Satellite
(1) Potential energy: 2
2
GMm LU mV
r mr
− −= = =
(2) Kinetic energy: 2
22
12 2 2
GMm LK mv
r mr= = =
(3) Total energy: 2
22 2 2GMm GMm GMm L
E U Kr r r mr
− − −= + = + = =
(4) Energy graph for a satellite (5) Binding Energy: The energy required to remove the
satellite from its orbit to infinity is called Binding Energy of
the system, i.e., Binding Energy (B.E.) 2
GMmE
r= − =
6.23 Weightlessness
The state of weightlessness (zero weight) can be observed in the following situations. (1) When objects fall freely under gravity (2) When a satellite revolves in its orbit around the earth (3) When bodies are at null points in outer space. The zero gravity region is called null point.
K
U E
O
+
–
En
erg
y
r
134 / Academic Excellence in Physics
7. Properties of Matter
7.1 Interatomic Forces
The forces between the atoms due to electrostatic interaction between the charges of the atoms are called interatomic forces.
(1) When two atoms are brought close to each other to a distance of the order of 10–10 m, attractive interatomic force is produced between two atoms.
(2) This attractive force increases continuously with decrease in r and becomes maximum for one value of r called critical distance, represented by x (as shown in the figure).
(3) When the distance between the two atoms becomes r0, the interatomic force will be zero. This distance r0 is called normal or equilibrium distance.
(4) When the distance between the two atoms further decreased, the interatomic force becomes repulsive in nature and increases very rapidly.
(5) The potential energy U is related with the interatomic force F by the following relation.
dU
Fdr
−=
When the distance between the two atoms becomes r0, the potential energy of the system of two atoms becomes minimum (i.e. attains maximum negative value hence the two atoms at separation r0 will be in a state of equilibrium.
7.2 Intermolecular Forces
The forces between the molecules due to electrostatic interaction between the charges of the molecules are called intermolecular forces. These forces are also called Vander Waal forces and are quite weak as compared to inter-atomic forces.
7.3 Comparison between Interatomic and Intermolecular Forces.
(i) Both the forces are electrical in origin. (ii) Both the forces are active over short distances. (iii) General shape of force-distance graph is similar for both the forces. (iv) Both the forces are attractive up to certain distance between atoms/molecules and
become repulsive when the distance between them become less than that value.
7.4 Solids.
A solid is that state of matter in which its constituent atoms or molecules are held strongly at the position of minimum potential energy and it has a definite shape and volume.
U
r O r0
Rep
uls
ion
F
r O r0
x
Attraction
Properties of Matter / 135
7.5 Elastic Property of Matter
(1) Elasticity: The property of matter by virtue of which a body tends to regain its original shape and size after the removal of deforming force is called elasticity.
(2) Plasticity: The property of matter by virtue of which it does not regain its original shape and size after the removal of deforming force is called plasticity.
(3) Perfectly elastic body: If on the removal of deforming forces the body regain its original configuration completely it is said to be perfectly elastic.
A quartz fibre and phosphor is the nearest approach to the perfectly elastic body. (4) Perfectly plastic body: If the body does not have any tendency to recover its original
configuration, on the removal of deforming force, it is said to be perfectly plastic. Paraffin wax, wet clay are the nearest approach to the perfectly plastic body. Practically there is no material which is either perfectly elastic or perfectly plastic. (5) Reason of elasticity: On applying the deforming forces, restoring forces are developed.
When the deforming force is removed, these restoring forces bring the molecules of the solid to their respective equilibrium position (r = r0) and hence the body regains its original form.
(6) Elastic limit: The maximum deforming force upto which a body retains its property of elasticity is called elastic limit of the material of body.
Elastic limit is the property of a body whereas elasticity is the property of material of the body.
(7) Elastic fatigue: The temporary loss of elastic properties because of the action of repeated alternating deforming force is called elastic fatigue.
It is due to this reason: (i) Bridges are declared unsafe after a long time of their use. (ii) Spring balances show wrong readings after they have been used for a long time. (iii) We are able to break the wire by repeated bending.
(8) Elastic after effect: The time delay in which the substance regains its original condition after the removal of deforming force is called elastic after effect. it is negligible for perfectly elastic substance, like quartz, phosphor bronze and large for glass fibre.
7.6 Stress
The internal restoring force acting per unit area of cross section of the deformed body is called stress.
Stress Force Area
F
A= =
Unit: N/m2 (S.I.), dyne/cm2 (C.G.S.) Stress developed in a body depends upon how the external forces are applied over it. On this basis there are two types of stresses: Normal and Shear or tangential stress (1) Normal stress: Here the force is applied normal to the surface. It is again of two types: Longitudinal and Bulk or volume stress
(i) Longitudinal stress
136 / Academic Excellence in Physics
(a) Deforming force is applied parallel to the length and causes increase in length. (b) Area taken for calculation of stress is area of cross section. (c) Longitudinal stress produced due to increase in length of a body under a deforming
force is called tensile stress. (d) Longitudinal stress produced due to decrease in length of a body under a
deforming force is called compressional stress.
(ii) Bulk or Volume stress (a) It occurs in solids, liquids or gases. (b) Deforming force is applied normal to surface at all points. (c) It is equal to change in pressure because change in pressure is responsible for
change in volume. (2) Shear or tangential stress: It comes in picture when successive layers of solid move on
each other i.e. when there is a relative displacement between various layers of solid. (i) Here deforming force is applied tangential to one of the faces. (ii) Area for calculation is the area of the face on which force is applied. (iii) It produces change in shape, volume remaining the same.
7.7 Strain.
The ratio of change in configuration to the original configuration is called strain. It has no dimensions and units. Strain are of three types:
(1) Linear strain: Change in length( )Linear strain
Original length( )l
l
∆=
Linear strain in the direction of deforming force is called longitudinal strain and in a direction perpendicular to force is called lateral strain.
(2) Volumetric strain: Change in volume( )Volumetric strain
Original volume( )V
V
∆=
(3) Shearing strain: It is defined as angle in radians through which a plane perpendicular to the fixed surface of the cubical body gets turned under the effect of tangential force.
x
Lφ =
� When a beam is bent both compression strain as well as an extension strain is produced.
7.8 Stress-strain Curve
(1) When the strain is small (region OP) stress is proportional to strain. This is the region where the so called Hooke’s law is obeyed. The point P is called limit of proportionality and slope of line OP gives the
F
Fixed face
φ φ
φ
L
x
Breaking
strength
Str
ess
Elastic
region
θ
Strain
P E A
B C
Plastic region
Elastic limit
O
Properties of Matter / 137
Young’s modulus Y of the material of the wire. Y = tanθ . (2) Point E known as elastic limit or yield-point. (3) Between EA, the strain increases much more. (4) The region EABC represents the plastic behaviour of the material of wire. (5) Stress-strain curve for different materials.
Brittle material Ductile material Elastomers
The plastic region between E and C is small for brittle material and it will break soon after the elastic limit is crossed.
The material of the wire have a good plastic range and such materials can be easily changed into different shapes and can be drawn into thin wires.
for elastomers and strain produced is much larger than the stress applied. Such materials have no plastic range and the breaking point lies very close to elastic limit. Example rubber.
7.9 Hooke’s law and Modulus of Elasticity.
According to this law, within the elastic limit, stress is proportional to the strain.
i.e. stress ∝ strain or stress
constantstrain
E= =
The constant E is called modulus of elasticity. (1) It’s value depends upon the nature of material of the body and the manner in which the
body is deformed. (2) It's value depends upon the temperature of the body. (3) It’s value is independent of the dimensions of the body. There are three modulii of elasticity namely Young’s modulus (Y), Bulk modulus (K) and
modulus of rigidity (η) corresponding to three types of the strain.
7.10 Young's Modulus (Y).
It is defined as the ratio of normal stress to longitudinal strain within limit of proportionality.
Normal stress /
longitudinal strain /F A FL
Yl L Al
= = =
Str
ess
Strain
C P
E
O
Str
ess
Strain
C P
E
O
Str
ess
Strain O
C
138 / Academic Excellence in Physics
Thermal stress: If a rod is fixed between two rigid supports, due to change in temperature its length will change and so it will exert a normal. This stress is called thermal stress. Thermal stress = Yα∆θ force produced in the body = YAα ∆θ
7.11 Work Done in Stretching a Wire.
In stretching a wire work is done against internal restoring forces. This work is stored in the wire as elastic potential energy or strain energy.
∴ Energy stored in wire 21 1
2 2YAl
U FlL
= =
Energy stored in per unit volume of wire.
2 21 1 1stress strain (strain) (stress)
2 2 2Y
Y= × × = × × =
7.12 Breaking of Wire.
When the wire is loaded beyond the elastic limit, then strain increases much more rapidly. The maximum stress corresponding to B (see stress-strain curve) after which the wire begin to flow and breaks, is called breaking stress or tensile strength and the force by application of which the wire breaks is called the breaking force.
(i) Breaking force depends upon the area of cross-section of the wire (ii) Breaking stress is a constant for a given (iii) Breaking force is independent of the length of wire. (iv) Breaking force ∝ πr2.
(v) Length of wire if it breaks by its own weight. PL
dg=
7.13 Bulk Modulus.
Then the ratio of normal stress to the volumetric strain within the elastic limits is called as Bulk modulus.
This is denoted by K.
Normal stress
volumetric strainK = ;
//
F A pVK
V V V
−= =
−∆ ∆
where p = increase in pressure; V = original volume; ∆V = change in volume The reciprocal of bulk modulus is called compressibility.
C = compressibility = 1 V
K pV
∆=
S.I. unit of compressibility is N–1m2 and C.G.S. unit is dyne–1 cm2. Gases have two bulk moduli, namely isothermal elasticity Eθ and adiabatic elasticity Eφ .
7.14 Density of Compressed Liquid
If a liquid of density ρ, volume V and bulk modulus K is compressed, then its density increases.
1P
K
∆ ′ρ = ρ + [1 ]C P= ρ + ∆ 1
As CK
=
Properties of Matter / 139
7.15 Modulus of Rigidity.
Within limits of proportionality, the ratio of tangential stress to the shearing strain is called modulus of rigidity of the material of the body and is denoted by η, i.e.
shear stress /shear strain
F A F
Aη = = =
φ φ
Only solids can exhibit a shearing as these have definite shape.
7.16 Poisson’s Ratio.
Lateral strain: The ratio of change in radius to the original radius is called lateral strain. Longitudinal strain: The ratio of change in length to the original length is called
longitudinal strain. The ratio of lateral strain to longitudinal strain is called Poisson’s ratio (σ).
i.e. Lateral strainLongitudinal strain
σ =
7.17 Relation Between Volumetric Strain, Lateral Strain and Poisson’s Ratio
11
2dV
AdL
σ = − [where A = cross-section of bar]
(i) If a material having σ = – 0.5 then Volume = constant i.e., the material is incompressible. (ii) Theoretical value of Poisson’s ratio –1 < σ < 0.5. (iii) Practical value of Poisson’s ratio 0 < σ < 0.5.
7.18 Relation between Y, k, η and σ.
Y = 3K(1 – 2σ) and Y = 2η(1 + σ); 93
KY
K
η=
+ η and 3 2
6 2K
K
− ησ =
+ η
7.19 Torsion of Cylinder
If the upper end of a cylinder is clamped and a torque is applied at the lower end the cylinder gets twisted by angle θ. Simultaneously shearing strain φ is produced in the cylinder.
(i) The angle of twist θ is directly proportional to the distance from the fixed end of the cylinder.
(ii) The value of angle of shear φ is directly proportional to the radius of the cylindrical shell.
(iii) Relation between angle of twist (θ) and angle of shear (φ)
AB = rθ = φl ∴ r
l
θφ =
(iv) Twisting couple per unit twist or torsional rigidity or torque required to produce unit twist.
4
2r
Cl
πη= ∴ C ∝ r4 ∝ A2
P r Q
φ
O
A B
θ
l
140 / Academic Excellence in Physics
(v) Work done in twisting the cylinder through an angle θ is 4 2
212 4
rW C
l
πη θ= θ =
7.20 Elastic Hysteresis.
Hysteresis loop: The area of the stress-strain curve is called the hysteresis loop and it is numerically equal to the work done in loading the material and then unloading it.
7.21 Factors Affecting Elasticity.
(1) Hammering and rolling: This result in increase in the elasticity of material. (2) Annealing: Annealing results in decrease in the elasticity of material. (3) Temperature: Elasticity decreases with rise in temperature but the elasticity of invar
steel (alloy) does not change with change of temperature. (4) Impurities: The type of effect depends upon the nature of impurities present in the
material.
7.22 Important Facts about Elasticity.
(1) The body which requires greater deforming force to produce a certain change in dimension is more elastic.
(2) When equal deforming force is applied on different bodies then the body which shows less deformation is more elastic. (i) Water is more elastic than air as volume change in water is less for same applied
pressure. (ii) Four identical balls of different materials are dropped from the same height then
after collision balls rises upto different heights. hivory > hsteel > hrubber > hclay because Yivory > Ysteel > Yrubber > Yclay.
(3) For a given material there can be different moduli of elasticity depending on the type of stress and strain.
(4) Ksolid > Kliquid > Kgas (5) Elasticity of a rigid body is infinite.
7.23 Practical Applications of Elasticity.
(i) The thickness of the metallic rope used in the crane is decided from the knowledge of elasticity.
(ii) Maximum height of a mountain on earth can be estimated. (iii) A hollow shaft is stronger than a solid shaft made of same mass, length and material.
7.24 Intermolecular Force.
The force of attraction or repulsion acting between the molecules are known as intermolecular force. The nature of intermolecular force is electromagnetic.
The intermolecular forces of attraction may be classified into two types.
Properties of Matter / 141
Cohesive force Adhesive force
The force of attraction between molecules of same substance is called the force of cohesion. This force is lesser in liquids and least in gases.
The force of attraction between the molecules of the different substances is called the force of adhesion.
7.25 Surface Tension
The property of a liquid due to which its free surface tries to have minimum surface area is called surface tension. A small liquid drop has spherical shape due to surface tension.Surface tension of a liquid is measured by the force acting per unit length on either side of an imaginary line drawn on the free surface of liquid. then T = (F/L).
(1) It depends only on the nature of liquid and is independent of the area of surface or length of line considered.
(2) It is a scalar as it has a unique direction which is not to be specified. (3) Dimension: [MT–2]. (Similar to force constant) (4) Units: N/m (S.I.) and Dyne/cm [C.G.S.]
7.26 Factors Affecting Surface Tension.
(1) Temperature: The surface tension of liquid decreases with rise of temperature Tt = T0(1 – αt)
where Tt, T0 are the surface tensions at t°C and 0°C respectively and α is the temperature coefficient of surface tension.
(2) Impurities: A highly soluble substance like sodium chloride when dissolved in water, increases the surface tension of water. But the sparingly soluble substances like phenol when dissolved in water, decreases the surface tension of water.
7.27 Surface Energy.
The potential energy of surface molecules per unit area of the surface is called surface energy. Unit: Joule/m2 (S.I.) erg/cm2 (C.G.S.) Dimension: [MT–2] ∴ W = T × ∆A [∆A = Total increase in area of the film from both the sides] i.e. surface tension may be defined as the amount of work done in increasing the area of the
liquid surface by unity against the force of surface tension at constant temperature.
7.28 Work Done in Blowing a Liquid Drop or Soap Bubble.
(1) If the initial radius of liquid drop is r1 and final radius of liquid drop is r2 then W = T × Increment in surface area
W = T × 4π ][ 21
22 rr − [drop has only one free surface]
(2) In case of soap bubble
W = T × 8π ][ 21
22 rr − [Bubble has two free surfaces]
142 / Academic Excellence in Physics
7.29 Splitting of Bigger Drop.
When a drop of radius R splits into n smaller drops, (each of radius r) then surface area of liquid increases. R3 = nr3
Work done = T × ∆A = T [Total final surface area of n drops – surface area of big drop] = T[n4πr2 – 4πR2].
7.30 Excess Pressure.
Excess pressure in different cases is given in the following table:
Plane surface Concave surface
∆P = 0
2TP
R∆ =
Convex surface Drop
2TP
R∆ =
2TP
R∆ =
Bubble in air Bubble in liquid
4TP
R∆ =
2TP
R∆ =
Bubble at depth h below the free surface
of liquid of density d
Cylindrical liquid surface
2TP hdg
R∆ = +
TP
R∆ =
∆P
∆P
∆P ∆P
∆P
h
∆P
∆ P = 0
R
Properties of Matter / 143
Liquid surface of unequal radii Liquid film of unequal radii
1 2
1 1P T
R R
∆ = +
1 2
1 12P T
R R
∆ = +
7.31 Shape of Liquid Meniscus
The curved surface of the liquid is called meniscus of the liquid.
If 2cF Fa=
tanα = ∞ ∴ α = 90o i.e. the resultant force acts vertically downwards. Hence the liquid meniscus must be horizontal.
2cF Fa<
tan α = positive ∴ α is acute angle i.e. the resultant force directed outside the liquid. Hence the liquid meniscus must be concave upward.
2cF Fa>
tan α = negative ∴ α is obtuse angle i.e. the resultant force directed inside the liquid. Hence the liquid meniscus must be convex upward.
Example: Pure water in silver coated capillary tube.
Example: Water in glass capillary tube.
Example: Mercury in glass capillary tube.
7.32 Angle of Contact
Angle of contact between a liquid and a solid is defined as the angle enclosed between the tangents to the liquid surface and the solid surface inside the liquid, both the tangents being drawn at the point of contact of the liquid with the solid.
θ < 90o;
2c
a
FF >
concave meniscus.
Liquid wets the solid surface.
θ = 90o; 2c
a
FF =
plane meniscus.
Liquid does not wet the solid surface.
θ > 90o; 2c
a
FF <
convex meniscus.
Liquid does not wet the solid surface.
(i) Its value lies between 0o and 180o θ = 0° for pure water and glass, θ = 90° for water and silver (ii) On increasing the temperature, angle of contact decreases.
Fa A
Fc α
FN
Fa A
Fc α
FN 45°
∆P ∆P
Fa A
Fc
α
FN
45°
θ θ θ
144 / Academic Excellence in Physics
(iii) Soluble impurities increases the angle of contact. (iv) Partially soluble impurities decreases the angle of contact.
7.33 Capillarity.
If a tube of very narrow bore (called capillary) is dipped in a liquid, it is found that the liquid in the capillary either ascends or descends relative to the surrounding liquid. This phenomenon is called capillarity.
The cause of capillarity is the difference in pressures on two sides curved surface of liquid.
7.34 Ascent Formula.
When one end of capillary tube of radius r is immersed into a liquid of density d which wets the sides of the capillary R = radius of curvature of liquid meniscus.
T = surface tension of liquid P = atmospheric pressure
∴ 2 cosTh
rdg
θ=
IMPORTANT POINTS
(i) The capillary rise depends on the nature of liquid and solid both i.e. on T, d, θ and R. (ii) Capillary action for various liquid-solid pair.
Meniscus Angle of contact Level
Concave θ < 90o Rises Plane θ = 90o No rise no fall
Convex θ > 90o Fall
(iii) Lesser the radius of capillary greater will be the rise and vice-versa. This is called Jurin’s law.
(iv) If the weight of the liquid contained in the meniscus is taken into consideration then
more accurate ascent formula is given by 2 cos3
T rh
rdg
θ= −
(v) In case of capillary of insufficient length, i.e., L < h, the liquid will neither overflow from the upper end. The liquid after reaching the upper end will increase the radius of its meniscus without changing nature such that:
hr = Lr′ ∵ L < h ∴ r ' > r
(vi) If a capillary tube is dipped into a liquid and tilted at an angle α from vertical, then the vertical height of liquid column remains same whereas the length of liquid column (l) in the capillary tube increases.
h = l cosα or cos
hl =
α
h α
R
Water
h l
Properties of Matter / 145
7.35 Pressure
The normal force exerted by liquid at rest on a given surface in contact with it is called thrust of liquid on that surface.
If F be the normal force acting on a surface of area A in contact with liquid, then pressure exerted by liquid on this surface is P = F/A AFP /=
(1) Units: N/m2 or Pascal (S.I.) and Dyne/cm2 (C.G.S.)
(2) Dimension: 2
1 22
[ ] [ ][ ] [ ]
[ ] [ ]F MLT
P ML TA L
−− −= = =
(3) Pressure is a tensor quantity. (4) Atmospheric pressure: 1 atm = 1/01 × 105 Pa = 1.01 bar = torr (5) If P0 is the atmospheric pressure then for a point at depth h below the surface of a liquid
of density ρ, hydrostatic pressure P is given by P = P0 + hρg. (6) Gauge pressure: The pressure difference between hydrostatic pressure P and
atmospheric pressure P0 is called gauge pressure. P – P0 + hρg
7.36 Density
In a fluid, at a point, density ρ is defined as: 0
limV
m dm
V dV∆ →
∆ρ = =
∆
(1) It has dimensions [ML–3] and S.I. unit kg/m3 while C.G.S. unit g/cc with 1 g/cc = 103kg/m3.
(2) Relative density or specific gravity which is defined as: Density of bodyDensity of water
RD =
(3) If m1 mass of liquid of density ρ1 and m2 mass of density ρ2 are mixed, then
1 2
1 1 2 2( / ) ( / ) ( / )i
i i
m m m m
V m m m p
+ ∑ρ = = =
ρ + ρ ∑
(4) With rise in temperature due to thermal expansion of a given body, volume will increase while mass will remain unchanged, so density will decrease, 0(1 – )ρ ρ γ∆θ≃
(5) With increase in pressure due to decrease in volume, density will increase.
0 1p
B
∆ ρ ρ +
≃ where B is blk modulus.
7.37 Pascal's Law
The increase in pressure at one point of the enclosed liquid in equilibrium of rest is transmitted equally to all other points of the liquid and also to the walls of the container, provided the effect of gravity is neglected.
Example: Hydraulic lift, hydraulic press and hydraulic brakes.
7.38 Archimedes Principle
When a body is immersed partly or wholly in a fluid, in rest it is buoyed up with a force equal to the weight of the fluid displaced by the body. This principle is called Archimedes principle.
146 / Academic Excellence in Physics
Apparent weight of the body of density ( ρ ) when immersed in a liquid of density (σ).
Apparent weight = Actual weight – Upthrust = W – Fup = Vρg – Vσg = V(ρ – σ)g = 1 –V gσ ρ ρ
∴ 1APPW W σ
= − ρ
(1) Relative density of a body (R.D.) = 1
1 2
Weight of body in airWeight in air–weight in water
W
W W=
−
(2) If the loss of weight of a body in water is 'a' while in liquid is 'b'
∴ air liquid
air water
Upthrust on body in liquid Loss of weight in liquidUpthrust on body in water Loss of weight in water
L
W
W Wa
b W W
−σ= = = =
σ −
7.39 Floatation.
(1) Translatory equilibrium: When a body of density ρ and volume V is immersed in a liquid of density ρ, the forces acting on the body are Weight of body W = mg = Vρg, acting vertically downwards through centre of gravity of the body.
Upthrust force = Vσg acting vertically upwards through the centre of gravity of the displaced liquid i.e., centre of buoyancy. (i) If density of body is greater than that of liquid ρ > σ
Weight will be more than upthrust so the body will sink (ii) If density of body is equal to that of liquid ρ = σ
Weight will be equal to upthrust so the body will float fully submerged in neutral equilibrium anywhere in the liquid.
(iii) If density of body is lesser than that of liquid ρ < σ Weight will be less than upthrust so the body will move upwards and in equilibrium
will float partially immersed in the liquid Vρg = Vinσg where Vin is the volume of body in the liquid.
(2) Rotatory Equilibrium: When a floating body is slightly tilted from equilibrium position, the centre of buoyancy B shifts. The vertical line passing through the new centre of buoyancy B′ and initial vertical line meet at a point M called meta-centre. If the meta-centre M is above the centre of gravity then object remains in stable equilibrium.However, if meta-centre goes below centre of gravity then object remains in stable equilibrium.
7.40 Streamline, Laminar and Turbulent Flow.
(1) Stream line flow: Stream line flow of a liquid is that flow in which each element of the liquid passing through a point travels along the same path and with the same velocity as the preceeding element passes through that point.
The two streamlines cannot cross each other and the greater is the crowding of streamlines at a place, the greater is the velocity of liquid particles at that place.
Properties of Matter / 147
(2) Laminar flow: If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different velocities which do not mix with each other, then the flow of liquid is called laminar flow.
In this flow the velocity of liquid flow is always less than the critical velocity of the liquid.
(3) Turbulent flow: When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of liquid becomes disordered or irregular. Such a flow is called a turbulent flow.
7.41 Critical Velocity and Reynold's Number.
The critical velocity is that velocity of liquid flow upto which its flow is streamlined and above which its flow becomes turbulent. Reynold's number is a pure number which determines the nature of flow of liquid through a pipe.
It is defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid.
So by the definition of Reynolds number 2Inertial force per unit area
Viscous force per unit area /R
v v rN
v r
ρ ρ= = =
η η
If the value of Reynold's number: (i) Lies between 0 to 2000, the flow of liquid is streamline or laminar. (ii) Lies between 2000 to 3000, the flow of liquid is unstable changing from streamline to
turbulent flow. (iii) Above 3000, the flow of liquid is definitely turbulent.
7.42 Equation of Continuity.
The equation of continuity is derived from the principle of conservation of mass.
For an incompressible, streamlined and non-viscous liquid product of area of cross section of tube and velocity of liquid remains constant.
i.e. a1v1 = a2v2
or av = constant; or 1
av
∝
When water falls from a tap, the velocity of falling water under the action of gravity will increase with distance from the tap (i.e, v2 > v1). So in accordance with continuity equation the cross section of the water stream will decrease (i.e., A2 < A1), i.e., the falling stream of water becomes narrower.
7.43 Energy of a Flowing Fluid
Pressure Energy Potential energy Kinetic energy
It is the energy possessed by a liquid by virtue of its pressure. It is the measure of work done in pushing the liquid against pressure
It is the energy possessed by liquid by virtue of its height or position above the surface of earth or any
It is the energy possessed by a liquid by virtue of its motion or velocity.
v
a
148 / Academic Excellence in Physics
without imparting any velocity to it.
reference level taken as zero level.
Pressure energy of the liquid PV
Potential energy of the liquid mgh
Kinetic energy of the liquid 212
mv
Pressure energy per unit
mass of the liquid P
ρ
Potential energy per unit mass of the liquid gh
Kinetic energy per unit mass of
the liquid 212
v
Pressure energy per unit volume of the liquid P
Potential energy per unit volume of the liquid ghρ
Kinetic energy per unit volume of
the liquid 212
vρ
7.44 Bernoulli's Theorem.
According to this theorem the total energy (pressure energy, potential energy and kinetic energy) per unit volume or mass of an incompressible and non-viscous fluid in steady flow through a pipe remains constant throughout the flow.
212
P gh v+ ρ + ρ = constant
(i) Bernoulli's theorem for unit mass of: 212
Pgh v+ + =
ρconstant
(ii) Dividing above equation by g we get 2
2P v
hg g
+ +ρ
= constant
Here P
gρ is called pressure head, h is called gravitational head and
2
2v
g is called velocity
head.
7.45 Applications of Bernoulli's Theorem
(i) Attraction between two closely parallel moving boats (ii) Working of an aeroplane: 'dynamic lift' (= pressure difference × area of wing) (iii) Action of atomiser: (iv) Blowing off roofs by wind storms (v) Magnus effect: When a spinning ball is thrown,
it deviates from its usual path in flight. This effect is called Magnus effect.
(vi) Venturimeter: It is a device used for measuring the rate of flow of liquid through pipes.
Rate of flow of liquid 22
21
21
2
aa
hgaaV
−=
C E
B
D
A
v1 v2
a2 a1
h
Properties of Matter / 149
7.46 Velocity of Efflux
Velocity of efflux from a hole made at a depth h below the free surface of the liquid(of depth H) is given by 2v gh= .
Which is same as the final speed of a free falling object from rest through a distance h. This result is known as Torricelli's theorem.
(i) Time taken by the liquid to reach the base-level 2( )H h
tg
−=
(ii) Horizontal range (x): 2 [2( ) / ] 2 ( )x vt gh H h g h H h= = × − = −
For maximum range 0dx
dh= ∴
2H
h =
∴ Maximum range max 22 2H H
x H H = − =
7.47 Viscosity and Newton's law of Viscous Force
The property of a fluid due to which it opposes the relative motion between its different layers is called viscosity (or fluid friction or internal friction) and the force between the layers opposing the relative motion is called viscous force.
Viscous force F is proportional to the area of the plane A and the velocity gradient dv
dx in a
direction normal to the layer,
i.e., dv
F Adx
= −η
Where η is a constant called the coefficient of viscosity. Negative sign is employed because viscous force acts in a direction opposite to the flow of liquid.
(1) Units: dyne-s-cm–2 or Poise (C.G.S. system); Newton-s-m–2 or Poiseuille or decapoise (S.I. system) 1 Poiseuille = 1 decapoise = 10 Poise (2) Dimension: [ML–1 T–1] (3) With increase in pressure, the viscosity of liquids (except water) increases while that of
gases is independent of pressure. The viscosity of water decreases with increase in pressure.
(4) Solid friction is independent of the area of surfaces in contact and the relative velocity between them.
(5) Viscosity represents transport of momentum, while diffusion and conduction represents transport of mass and energy respectively.
(6) The viscosity of gases increases with increase of temperature. (7) The viscosity of liquid decreases with increase of temperature.
7.48 Stoke's Law and Terminal Velocity
150 / Academic Excellence in Physics
Stokes established that if a sphere of radius r moves with velocity v through a fluid of viscosity η, the viscous force opposing the motion of the sphere is
F = 6πηrv (Stokes law) If a spherical body of radius r is dropped in a viscous
fluid, it is first accelerated and then it's acceleration becomes zero and it attains a constant velocity called terminal velocity.
Terminal velocity 22 ( )
9r g
vρ − σ
=η
(i) If ρ > σ then body will attain constant velocity in downward direction. (ii) If ρ < σ then body will attain constant velocity in upward direction. Example: Air bubble in a liquid and clouds in sky. (iii) Terminal velocity graph:
7.49 Poiseuille’s Formula.
Poiseuille studied the stream-line flow of liquid in capillary tubes. He found that if a pressure difference (P) is maintained across the two ends of a capillary tube of length 'l ' and radius r, then the volume of liquid coming out of the tube per second is
4
8P r
Vl
π=
η (Poiseulle's equation)
This equation also can be written as, P
VR
= where 4
8 lR
r
η=
π
R is called as liquid resistance. (1) Series combination of tubes:
(i) The volume of liquid flowing through both the tubes i.e. rate of flow of liquid is same.
(ii) Effective liquid resistance in series combination Reff = R1 + R2 (2) Parallel combination of tubes:
(i) Pressure difference across both tubes remains same.
(ii) Effective liquid resistance in parallel combination 1 2
1 1 1
effR R R= +
VT
V
Time or distance (R)
Simple Harmonic Motion / 151
9. Simple Harmonic Motion
9.1 Periodic Motion
A motion, which repeat itself over and over again after a regular interval of time is called a periodic motion and the fixed interval of time after which the motion is repeated is called period of the motion.
Examples: Revolution of earth around the sun (period one year).
9.2 Oscillatory or Vibratory Motion.
The motion in which a body moves to and fro or back and forth repeatedly about a fixed point in a definite interval of time .Oscillatory motion is also called as harmonic motion.
Example: The motion of the pendulum of a wall clock.
9.3 Harmonic and Non-harmonic Oscillation.
Harmonic oscillation is that oscillation which can be expressed in terms of single harmonic function (i.e. sine or cosine function). Example: y = α sin ωt or y = a cos ωt.
Non-harmonic oscillation is that oscillation which can not be expressed in terms of single harmonic function. Example: y = a sin ωt + b sin 2 ωt.
9.4 Some Important Definitions.
(1) Time period: It is the least interval of time after which the periodic motion of a body repeats itself. S.I. units of time period is second.
(2) Frequency: It is defined as the number of periodic motions executed by body per second. S.I unit of frequency is hertz (Hz).
(3) Angular Frequency: ω = 2 πn (4) Displacement: Its deviation from the mean position. (5) Phase: It is a physical quantity, which completely express the position and direction of
motion, of the particle at that instant with respect to its mean position. 0sin sin( )y a a t= θ = ω + φ here, θ = ωt + φ0 = phase of vibrating particle.
(i) Initial phase or epoch: It is the phase of a vibrating particle at t = 0. (ii) Same phase: Two vibrating particle are said to be in same phase, if the phase
difference between them is an even multiple of π or path difference is an even multiple of (λ / 2) or time interval is an even multiple of (T/2).
(iii) Opposite phase: Opposite phase means the phase difference between the particle is an odd multiple of π or the path difference is an odd multiple of λ or the time interval is an odd multiple of (T / 2).
(iv) Phase difference: If two particles performs S.H.M and their equation are y1 = a sin(ωt + φ1) and y2 = a sin(ωt + φ2) then phase difference ∆φ = (ωt + φ2) – (ωt + φ1) = φ2 – φ1
9.5 Simple Harmonic Motion.
Simple harmonic motion is a special type of periodic motion, in which Restoring force ∝ Displacement of the particle from mean position.
152 / Academic Excellence in Physics
F = – kx Where k is known as force constant. Its S.I. unit is Newton/meter and dimension is [MT –2].
9.6 Displacement in S.H.M.
Simple harmonic motion is defined as the projection of uniform circular motion on any diameter of circle of reference
(i) y = a sin ωt when at t = 0 the vibrating particle is at mean position. (ii) y = a cos ωt when at t = 0 the the vibrating particle is at extreme position. (iii) y = a sin(ωt ± φ) when the vibrating particle is φ phase leading or lagging from the
mean position.
9.7 Comparative Study of Displacement, Velocity and Acceleration
Displacement y = a sin ωt Velocity v = aω cos
ωt = a sin2
tπ ω ω +
Acceleration A = –aω2 sin ωt = aω2 sin(ωt + π) (i) All the three quantities
displacement, velocity and acceleration show harmonic variation with time having same period.
(ii) The velocity amplitude is ω times the displacement amplitude
(iii) The acceleration amplitude is ω2 times the displacement amplitude (iv) In S.H.M. the velocity is ahead of displacement by a phase angle π/2. (v) In S.H.M. the acceleration is ahead of velocity by a phase angle π/2. (vi) The acceleration is ahead of displacement by a phase angle of π (vii) Various physical quantities in S.H.M. at different position:
Physical quantities Equilibrium position (y =
0)
Extreme Position (y = ± a)
Displacement y = a sin ωt Minimum (Zero) Maximum (a)
Velocity 22 yav −= ω Maximum (aω) Minimum (Zero)
Acceleration 2A yω= − Minimum (Zero) Maximum ( a2ω )
T T
2
1
T2
3– aω2
2T Time
2T T
2T T
2T T
0
0
0
a
+aω2
– aω
v
y
– a
+aω
+a
Displacement
y = a sin ωt
Velocity
v = aω cos ωt
Acceleration
A = – aω2 cos ωt
Simple Harmonic Motion / 153
9.8 Energy in S.H.M.
A particle executing S.H.M. possesses two types of energy: Potential energy and Kinetic energy
(1) Potential energy: 2 2 21sin
2U m a t= ω ω
(i) 2 2 2max
1 12 2
U ka m a= = ω when y = ± a; ωt = π/2; t = T/4
(ii) Umin = 0 when y = 0; ωt = 0; t = 0. (2) Kinetic energy:
2 2 21cos
2K ma t= ω ω 2 2 21
( )2
K m a y= ω −
(i) max
2 212
K m a= ω when y = 0; t = 0; ωt = 0
(ii) Kmin = 0 when y = a; t = T/4, ωt = π/2 (3) Total energy: Total mechanical energy = Kinetic
energy + Potential energy
2 212
E m a= ω
Total energy is not a position function i.e. it always remains constant.
(4) Energy position graph: (5) Kinetic energy and potential energy vary
periodically with double the frequency of S.H.M
9.9 Time Period and Frequency of S.H.M.
Time period 2( ) 2
mT
k
π= = π
ω as k
mω =
Frequency (n) 1 12
k
T m= =
π
In general m is called inertia factor and k is called spring factor.
Thus Inertia factor
2Spring factor
T = π
9.10 Differential Equation of S.H.M.
For S.H.M. (linear) 2
2 0d y
m kydt
+ = [As k
mω = ]
For angular S.H.M. 2
2 02
d
dt
θ+ ω θ = [ 2 c
Iω = ]
y =– a y =+ a y = 0
U
Energy
K
154 / Academic Excellence in Physics
9.11 Simple Pendulum
Mass of the bob = m
Effective length of simple pendulum = l; 2l
Tg
= π
(i) The period of simple pendulum is independent of amplitude as long as its motion is simple harmonic.
(ii) Time period of simple pendulum is also independent of mass of the bob. (iii) If the length of the pendulum is comparable to the radius of earth then
12
1 1T
gl R
= π +
. If ( )1 / 1 /l R l R>> → ∞ < so 2 84.6R
Tg
= π ≅ minutes
(iv) The time period of simple pendulum whose point of suspension moving horizontally with
acceleration a 2 2 1 / 22( )
lT
g a= π
+ and 1tan ( / )a g−θ =
(v) Second’s Pendulum: It is that simple pendulum whose time period of vibrations is two seconds.
(vi) Work done in giving an angular displacement θ to the pendulum from its mean position. W = U = mgl(1 – cos θ) (vii) Kinetic energy of the bob at mean position = work done or potential energy at extreme (viii) Various graph for simple pendulum.
l ∝ T2
T
l
l ∝ T2
T2
l
l ∝ T2
T
l
T ∝g
1
T
g
1
T
g
T ∝g
1
Simple Harmonic Motion / 155
9.12 Spring Pendulum
A point mass suspended from a mass less spring or placed on a frictionless horizontal plane attached with spring constitutes a linear harmonic spring pendulum
Time period inertia factor
2spring factor
T = π
2m
Tk
= π and Frequency 12
kn
m=
π
(i) Time of a spring pendulum is independent of acceleration due to gravity. (ii) If the spring has a mass M and mass m is suspended from it, effective mass is given by
3e
Mm m
ff= +
So that 2 em
ffT
k= π
(v) If two masses of mass m1 and m2 are connected by a spring and made to oscillate on
horizontal surface, the reduced mass mr is given by 1 2
1 1 1
rm m m= +
So that 2 rmT
k= π
(vi) If a spring pendulum, oscillating in a vertical plane is made to oscillate on a horizontal surface, (or on inclined plane) time period will remain unchanged.
(vii) If the stretch in a vertically loaded spring is 0y then
02 2m y
Tk g
= π = π
` Time period does not depends on ‘g’ because along with g, yo will also change in such a
way that 0y m
g k= remains constant.
(viii) Series combination: If n springs of different force constant are connected in series having
force constant 1 2 3, , .......k k k respectively then 1 2 3
1 1 1 1........
ek k k k
ff
= + + +
(ix) Parallel combination: If the springs are connected in parallel then 1 2 3k k k keff
= + + + ….
(x) If the spring of force constant k is divided in to n equal parts then spring constant of each part will become nk .
(xi) The spring constant k is inversely proportional to the spring length.
As 1 1Extension Lengthof spring
k ∝ ∝
k
m2 m1
156 / Academic Excellence in Physics
(xii) When a spring of length l is cut in two pieces of length l1 and l2 such that l1 = nl2.
If the constant of a spring is k then Spring constant of first part 1
( 1)k nk
n
+=
Spring constant of second part k2 = (n + 1)k and ratio of spring constant 1
2
1k
k n=
9.13 Various Formulae of S.H.M..
S.H.M. of a liquid in U tube
If a liquid of density ρ contained in a vertical U tube performs S.H.M. in its two
limbs. Then time period 2 22L h
Tg g
= π = π
where L = Total length of liquid column, h = Height of undisturbed liquid in each
limb (L=2h)
S.H.M. of a bar magnet in a magnetic field
2I
TMB
= π
I = Moment of inertia of magnet
M = Magnetic moment of magnet
B = Magnetic field intensity
S.H.M. of a floating cylinder
If l is the length of cylinder dipping in
liquid then time period 2l
Tg
= π
S.H.M. of ball in the neck of an air
chamber
2 mVT
A E
π=
m = mass of the ball
V = volume of air- chamber
A = area of cross section of neck
E = Bulk modulus for Air
S.H.M. of a small ball rolling down in
hemi-spherical bowl
2R r
Tg
−= π
R = radius of the bowl
r =radius of the ball
S.H.M. of a body suspended from a wire
2mL
TYA
= π
m = mass of the body
L = length of the wire
Y = young’s modulus of wire
A = area of cross section of wire
l
R
m
L
h
N
S F
F
Simple Harmonic Motion / 157
S.H.M. of a piston in a cylinder
2Mh
TPA
= π
M = mass of the piston A = area of cross section h = height of cylinder P = pressure in a cylinder
S.H.M of a cubical block
2M
TL
= πη
M = mass of the block L = length of side of cube η = modulus of rigidity
S.H.M. of a body in a tunnel dug along any chord of earth
2R
Tg
= π = 84.6 minutes
S.H.M. of body in the tunnel dug along the diameter of earth
2R
Tg
= π
T = 84.6 minutes R = radius of the earth = 6400km g = acceleration due to gravity = 9.8m/s2 at earth’s surface
S.H.M. of a conical pendulum
cos2
LT
g
θ= π
L = length of string
θ = angle of string from the vertical
g = acceleration due to gravity
S.H.M. of L-C circuit
2T LC= π
L = coefficient of self inductance
C = capacity of condenser
9.14 Important Facts and Formulae
(1) When a body is suspended from two light springs separately. The time period of vertical oscillations are T1 and T2 respectively.
When these two springs are connected in series and the same mass m is attached at lower end and then
Time period of the system 2 21 2T T T= +
When these two springs are connected in parallel and the same mass m is attached at lower end then
Time period of the system 1 2
2 21 2
T TT
T T=
+
(2) If infinite spring with force constant k, 2k, 4k, 8k ……. respectively are connected in series. The effective force constant of the spring will be k/2.
L
θ
R R
O
L T
θ
158 / Academic Excellence in Physics
(4) If y1 = a sin ωt and y2 = b cos ωt are two S.H.M. then by the superimposition of these two S.H.M. we get a S.H.M. as y = A sin(ωt + φ)
where 2 2A a b= + and 1tan ( / )b a−φ =
9.15 Free, Damped, Forced and Maintained Oscillation.
(1) Free oscillation (i) The oscillation of a particle with fundamental frequency under the influence of
restoring force are defined as free oscillations (ii) The amplitude, frequency and energy of oscillation remains constant (iii) Frequency of free oscillation is called natural frequency.
(2) Damped oscillation (i) The oscillation of a body whose amplitude goes on decreasing with time are defined
as damped oscillation (ii) Amplitude of oscillation decreases exponentially due to damping forces like
frictional force, viscous force, hystersis etc. (3) Forced oscillation
(i) The oscillation in which a body oscillates under the influence of an external periodic force are known as forced oscillation
(ii) Resonance: When the frequency of external force is equal to the natural frequency of the oscillator. Then this state is known as the state of resonance. And this frequency is known as resonant frequency.
(4) Maintained oscillation: The oscillation in which the loss of oscillator is compensated by the supplying energy from an external source are known as maintained oscillation.
Wave Motion / 159
10. Wave Motion
10.1 Wave
A wave is a disturbance which propagates energy and momentum from one place to the other without the transport of matter.
(1) Necessary properties of the medium for wave propagation: (i) Elasticity: So that particles can return to their mean position, after having been disturbed. (ii) Inertia: So that particles can store energy and overshoot their mean position. (iii) Minimum friction amongst the particles of the medium. (iv) Uniform density of the medium.
(2) Mechanical waves: The waves which require medium for their propagation are called mechanical waves.
Example: Waves on string and spring, waves on water surface, sound waves, seismic waves. (3) Non-mechanical waves: The waves which do not require medium for their propagation
are called non- mechanical or electromagnetic waves. Examples: Light, heat (Infrared), radio waves, γ- rays, X-rays etc. (4) Transverse waves: Particles of the medium execute simple harmonic motion about their
mean position in a direction perpendicular to the direction of propagation of wave motion. (i) It travels in the form of crests and troughs. (ii) A crest is a portion of the medium which is raised temporarily. (iii) A trough is a portion of the medium which is depressed temporarily. (iv) Examples of transverse wave motion: Movement of string of a sitar, waves on the
surface of water. (v) Transverse waves can not be transmitted into liquids and gases.
(5) Longitudinal waves: If the particles of a medium vibrate in the direction of wave motion the wave is called longitudinal. (i) It travels in the form of compression and rarefaction. (ii) A compression (C) is a region of the medium in which particles are compressed. (iii) A rarefaction (R) is a region of the medium in which particles are rarefied. (iv) Examples sound waves travel through air in the form of longitudinal waves. (v) These waves can be transmitted through solids, liquids and gases.
10.2 Important Terms
(1) Wavelength:
(i) It is the length of one wave. (ii) Distance travelled by the wave in one time period is known as wavelength.
λ = Distance between two consecutive crests or troughs. (2) Frequency: Number of vibrations completed in one second. (3) Time period: Time period of vibration of particle is defined as the time taken by the
particle to complete one vibration about its mean position.
160 / Academic Excellence in Physics
(4) Relation between frequency and time period: Time period = 1/Frequency ⇒ T = 1/n (5) Relation between velocity, frequency and wavelength: v = nλ
10.3 Sound Waves
The energy to which the human ears are sensitive is known as sound. According to their frequencies, waves are divided into three categories: (1) Audible or sound waves: Range 20 Hz to 20 KHz. (2) Infrasonic waves: Frequency lie below 20 Hz. (3) Ultrasonic waves: Frequency greater than 20 KHz.
� Supersonic speed: An object moving with a speed greater than the speed of sound is said to move with a supersonic speed.
� Mach number: It is the ratio of velocity of source to the velocity of sound.
Mach Number = Velocity of sourceVelocity of sound
.
10.4 Velocity of Sound (Wave motion)
(1) Speed of transverse wave motion:
(i) On a stretched string: Tv
m= T = Tension in the string; m = Linear density of string
(mass per unit length).
(ii) In a solid body: vη
=ρ
η = Modulus of rigidity; ρ = Density of the material.
(2) Speed of longitudinal wave motion:
(i) In a solid long bar Y
v=ρ
Y = Young's modulus; ρ = Density
(ii) In a liquid medium k
v=ρ
k = Bulk modulus
(iii) In gases k
v=ρ
10.5 Velocity of Sound in Elastic Medium
Velocity of sound in any medium is E
v=ρ
(E = Elasticity of the medium; ρ = Density of the medium)
(1) vsteel > vwater > vair 5000 m/s > 1500 m/s > 330 m/s (2) Newton's formula: He assumed that propagation of sound is isothermal
vair = K P
=ρ ρ
As K = Eθ = P ; Eθ = Isothermal elasticity; P = Pressure.
By calculation vair = 279 m/sec. However the experimental value of sound in air is 332 m/sec
Wave Motion / 161
(3) Laplace correction: He modified that propagation of sound in air is adiabatic process.
v = Ek φ=
ρ ρ (As k = Eφ = γρ = Adiabatic elasticity)
v = 331.3 m/s (γAir = 1.41)
(4) Effect of density: v = Pγ
ρ⇒
1v ∝
ρ
(5) Effect of pressure: Velocity of sound is independent of the pressure (when T = constant)
(6) Effect of temperature: ( )inv T K∝
When the temperature change is small then vt = v0 (1 + αt)
Value of α = 0.608 /
o
m s
C= 0.61 (Approx.)
(7) Effect of humidity: With rise in humidity velocity of sound increases. (8) Sound of any frequency or wavelength travels through a given medium with the same velocity.
10.6 Reflection of Mechanical Waves
Medium Longitudinal
wave
Transverse
wave
Change
in
direction
Phase
change
Time
change
Path
change
Reflection from rigid end/denser medium
Compression as rarefaction and vice-versa
Crest as crest and Trough as trough
Reversed π 2T
2λ
Reflection from free end/rarer medium
Compression as compression and rarefaction as rarefaction
Crest as trough and trough as crest
No change Zero Zero Zero
10.7 Progressive Wave
(1) These waves propagate in the forward direction of medium with a finite velocity. (2) Energy and momentum are transmitted in the direction of propagation of waves. (3) In progressive waves, equal changes in pressure and density occurs at all points of
medium. (4) Various forms of progressive wave function.
(i) y = A sin (ω t – kx)
(ii) y = A sin 2–t x
π ω λ
where y = displacement A = amplitude ω = angular frequency n = frequency k = propagation constant T = time period λ = wave length v = wave velocity t = instantaneous time x = position of particle from origin
162 / Academic Excellence in Physics
(iii) y = A sin 2t x
T
π − λ
(iv) y = A sin 2π
λ (vt – x)
(v) y = A sin ω x
tv
−
(a) If the sign between t and x terms is negative the wave is propagating along positive X-axis and if the sign is positive then the wave moves in negative X-axis direction.
(b) The Argument of sin or cos function i.e. (ωt – kx) = Phase.
(c) The coefficient of t gives angular frequency ω= 2
2 nT
ππ = = vk.
(d) The coefficient of x gives propagation constant or wave number =k2
v
π ω=
λ.
(e) The ratio of coefficient of t to that of x gives wave or phase velocity. i.e. vk
ω= .
(f) When a given wave passes from one medium to another its frequency does not change.
(g) From v = nλ ⇒ v ∝ λ ∵ n = constant ⇒ 1 1
2 2
v
v
λ=
λ.
(5) Some terms related to progressive waves
(i) Wave number ( n ): The number of waves present in unit length. ( n ) = 1λ
.
(ii) Propagation constant (k): k = x
φ
Angular velocityWave velocity
kv
ω= = and
22k
π= = πλ
λ
(iii) Wave velocity (v): v = 2
nk T
ω ωλ λ= λ = =
π.
(iv) Phase and phase difference 2
( , ) ( )x t vt xπ
φ = −λ
.
(v) Phase difference =2T
π× Time difference.
(vi) Phase difference = 2π
×λ
Path difference
⇒ Time difference = T
×λ
Path difference.
Wave Motion / 163
10.8 Principle of Superposition
If 1 2, 3,y y y� � � ………. are the displacements at a particular time at a particular position, due to
individual waves, then the resultant displacement. 1 2 3 .............y y y y= + + +� � � �
Important applications of superposition principle: (a) Stationary waves; (b) Beats.
10.9 Standing Waves or Stationary Waves
When two sets of progressive wave trains of same type (both longitudinal or both transverse) having the same amplitude and same time period/frequency/wavelength travelling with same speed along the same straight line in opposite directions superimpose, a new set of waves are formed. These are called stationary waves or standing waves.
Characteristics of standing waves: (1) The disturbance confined to a particular region (2) There is no forward motion of the disturbance beyond this particular region. (3) The total energy is twice the energy of each wave. (4) Points of zero ampitude are known as nodes.
The distance between two consecutive nodes is 2λ
(5) Points of maximum amplitude is known as antinodes. The distance between two consecutive antinodes is also λ/2. The distance between a node and adjoining antinode is λ/4.
(6) The medium splits up into a number of segments. (7) All the particles in one segment vibrate in the same phase. Particles in two consecutive
segments differ in phase by 180°. (8) Twice during each vibration, all the particles of the medium pass simultaneously through
their mean position.
10.10 Standing Waves on a String
Incident wave y1 = a sin 2π
λ (vt + x)
Reflected wave ( )2
2siny a vt x
π= − + π λ
Using superposition principle: y = y1 + y2 = 2 a
cos 2 2
sinvt xπ π
λ λ
1. First normal mode of vibration:
x = 0 x = L
y1
y2
N N
2
1λ=L
A
164 / Academic Excellence in Physics
2. Second normal mode of vibration:
3. Third normal mode of vibration:
10.11 Standing Wave in a Closed Organ Pipe
Equation of standing wave y = 2a cos 2 2
sinvt xπ π
λ λ
1. First normal mode of vibration:
2. Second normal mode of vibration:
3. Third normal mode of vibration:
N N A A
N
L = λ2
N
N
N
N A A A
N
2
1λ=L
A
N A N
A
N A
N A A N
4
3 2λ=L
Wave Motion / 165
10.12 Standing Waves in Open Organ Pipes.
(1) First normal mode of vibration:
(2) Second normal mode of vibration
(3) Third normal mode of vibration
10.13 Comparative Study of Stretched Strings, Open Organ Pipe and Closed Organ Pipe
S.
No.
Parameter Stretched
string
Open organ pipe Closed organ
pipe
(1) Fundamental frequency or 1st harmonic(1st mode of vibration)
1 2v
nl
= 1 2v
nl
= 1 4v
nl
=
(2) Frequency of 1st overtone or 2nd harmonic(2nd mode of vibration)
n2 = 2n1 n2 = 2n1 Missing
(3) Frequency of 2nd overtone or 3rd harmonic((3rd mode of vibration)
n3 = 3n1 n3 = 3n1 n3 = 3n1
(4) Frequency ratio of overtones
2: 3: 4… 2: 3: 4… 3: 5: 7…
L = λ2
N
A
N
A A
2
3 3λ=L
N A
N A A
N
A
A A N
2
1λ=L
166 / Academic Excellence in Physics
(5) Frequency ratio of harmonics
1: 2: 3: 4… 1: 2: 3: 4… 1: 3: 5: 7…
(6) Nature of waves Transverse stationary
Longitudinal stationary
Longitudinal stationary
(7) General formula for wavelength
2L
nλ = n = 1,2,3,
…
2=
L
nλ n = 1,2,3, …
( )4
2 1=
−
L
nλ
(8) Position of nodes n
L
n
L
n
Lx ..........
3,
2,,0=
( )3 5 2 1, , ....
2 2 2 2
L L L Lx
n n n n
η −=
2 4 6 2 = 0, , , .........
(2 1) (2 1) (2 1) (2 1)
L L L nLx
n n n n− − − −
(9) Position of antinodes
( )3 5 2 1, , ....
2 2 2 2
L L L Lx
n n n n
η −=
n
L
n
L
n
Lx ............
3,
2,,0=
3 5, , .........,
2 1 2 1 2 1
L L Lx L
n n n=
− − −
(i) Harmonics are the notes/sounds of frequency equal to or an integral multiple of
fundamental frequency (n). (ii) Overtones are the notes/sounds of frequency twice/thrice/ four times the fundamental
frequency (n). (iii) In organ pipe an antinode is not formed exactly at the open end rather it is formed a
little distance away from the open end outside it. The distance of antinode from the open end of the pipe is = 0.6r (where r is radius of organ pipe). This is known as end correction.
10.14 Vibration of a String
General formula of frequency 2p
p Tn
L m=
L = Length of string, T = Tension in the string m = Mass per unit length (linear density), p = mode of vibration (1) The string will be in resonance with the given body if any of its natural frequencies
concides with the body.
(2) If M is the mass of the string of length L, M
mL
= .
So 1
2T
nLr
=πρ
(r = Radius, ρ = Density)
10.15 Beats
When two sound waves of slightly different frequencies, travelling in a medium along the same direction, superimpose on each other, the intensity of the resultant sound at a particular position rises and falls regularly with time. This phenomenon is called beats.
(1) Beat period: The time interval between two successive beats (i.e. two successive maxima of sound) is called beat period.
(2) Beat frequency: The number of beats produced per second is called beat frequency.
Wave Motion / 167
(3) Persistence of hearing: The impression of sound heard by our ears persist in our mind for 1/10th of a second.
So for the formation of distinct beats, frequencies of two sources of sound should be nearly equal (difference of frequencies less than 10)
(4) Equation of beats: If two waves of equal amplitudes 'a' and slightly different frequencies n1 and n2 travelling in a medium in the same direction then equation of beats is given by
y = A sin π(n1 – n2)t where A = 2a cos π(n1 – n2)t = Amplitude of resultant wave. (5) Beat frequency: n = n1 ~ n2.
(6) Beat period: T=1 2
1 1Beat frequency ~n n
=
10.16 Doppler Effect
Whenever there is a relative motion between a source of sound and the listener, the apparent frequency of sound heard by the listener is different from the actual frequency of sound emitted by the source.
Apparent frequency n' =( )( )
m L
m S
v v v n
v v v
+ − + −
Here n = Actual frequency; vL = Velocity of listener; vS = Velocity of source vm = Velocity of medium and v = Velocity of sound wave Sign convention: All velocities along the direction S to L are taken as positive and all
velocities along the direction L to S are taken as negative. If the medium is stationary vm = 0
then n' = L
S
v vn
v v
−
−
(1) No Doppler effect takes place(n' = n) when relative motion between source and listener is zero.
(2) Source and listener moves at right angle to the direction of wave propagation.(n' = n) (i) If the velocity of source and listener is equal to or greater than the sound velocity
then Doppler effect is not observed. (ii) Doppler effect does not says about intensity of sound. (iii) Doppler effect in sound is asymmetric but in light it is symmetric.
10.17 Some Typical Features of Doppler’s Effect in Sound.
(1) When a listener moves between two distant sound sources: In this case listener observed beats. Let vL be the velocity of listener away from S1 and towards S2. Frequency of both the sources = n, velocity of sound = v then
∴ Beat frequency 2 Lnv
v=
168 / Academic Excellence in Physics
(2) When source is revolving in a circle and listener L is on one side
maxs
nvn
v v=
− and min
s
nvn
v v=
+
(3) When listener L is moving in a circle and the source is on one side
max
( )Lv v nn
v
+= and min
( )Lv v nn
v
−=
169
Life of Science Articles
The science teachers have been facing great difficulty in maintaining the stock of science materials.Majority of them do not know the life of various science equipments. To help, the teachers, the“Science Branch News Bulletin” has reproduced the copy of the circular issued by the Directorof Education order No. F-4 (52)/-61 Edn (P) dated 2nd August, 1962 which is as follows :
The Director of Education is pleased to accept the recommendation of the Committee, appointedby him, to fix the life of furniture articles and other non-consumable articles (science equipmentsetc.) used in the Directorate of Education, Delhi/and the Govt./Aided schools in the Union Territoryof Delhi, vide order no. F(52)/61-Edn({) dated 1.9.1961 and accordingly fixes the life of eachitem, as shown in the Annexures “A”, B, C, D, E, F, H and I to the extent mentioned therein.
2. In case there is any other item, the life of whih has not been fixed, the same may kindly be intimatedto this Branch urgently, so that the same may also be considered.
3. In the case of schools, which are/were in tents, the life of the articles, to be condemned, maybe reduced by 25 p.c. of the life, as fixed in the attached Annexures, provided the condemnationBoard are satisfied for the proper use of the articles in question. A certificate to this effect mustbe endorsed on he lists. (This para was subsequently added vide letter no. F.4(52)/61-Edn(P), datedDecember, 1962).
Annexure ‘A’ (Years)
1. Almirah wooden 20
2. Almirah Iron 50
3. Black Board 3
4. Black Board Stand 3
5. Benches 5
6. Buckets (Tin) 2
7. Bicycle 8
8. Bicycles stand (iron) 10
9. Bicycle stand (wooden) 3
10. Chairs wooden seat 5
11. Chairs iron/steel seat 10
12. Chairs cane seat 5
13. Cash Box (wooden) 5
14. Cash Box (iron) 25
15. Chauki (Takhat) 10
16. Desk Single Shift 7
17. Desk Double Shift 5
18. Stool 3
19. Durries 5
20. Rack (wooden) 10
21. Rack (iron) 25
22. Officers Table 20
APPENDIX
170
23. Teachers Table 7
24. Office Table 10
25. Library Table 10
26. Physics Table 10
27. Chemistry Table 8
28. Domestic Science Table 8
29. Biology Table 10
30. Table Cloth 2
31. Screens 10
32. Newspaper Stand 5
33. Waste Paper Basket (Tin) 5
34. Waste Paper Basket (wooden) 3
35. Notice Board 5
36. Tray (wooden) 5
37. Tray (iron) 5
38. Paper Stand 7
39. Foot Rest 5
40. Hat Hanger & Looking glass 10
41. Confidential Box 10
42. Teapoy wooden 10
43. Carpet 10
44. Trunk 10
45. Map Stand 5
46. Cash Safe 50
1. Balance (Spring) 5
2. Balance (Physical) 5
3. Weight Boxes 5
4. Boyle’s Law Apparatus 5
5. Vernier Callipers 5
6. Fortin’s Barometer 5
7. Metallic Cylinder 5
8. Metal Sphere 7
9. Metres rod (wooden) 1
10. S.G. Bottle 5
11. Spherometer 5
12. Screw Gauge 5
13. StopWatch 10
14. Inclined plane 5
15. Gravesand’s apparatus 5
16. Young’s modulus 5
17. Concave Mirror 2
18. Convex Lens 5
19. Glass Prism 5
20. Glass Slab 5
21. Screen (Glass) 2
22. Lens Stand (wooden) 2
23. Optical bench (wooden) 2
24. Spectomdeter 5
25. Wire Gauge Stand Iron 5
26. -do-Wooden 3
27. Travelling Microscope 10
28. Copper Calorimeter 5
29. Hypsometer (Copper) 5
30. Thermometer 1
31. Max. Min. Thermometer 5
32. Magnet (bar) 5
33. Compass needle 3
34. Compass (for lines of force) 5
35. Deflection Mangnetometer 5
36. Ammeter 10
37. Voltameter 10
38. Galvanometer 7
39. Accumulator 2
40. Laclanche Cell 2
Annexure ‘B’ Physics Apparatus (Non-consumable)
171
41. Electric Bell 2
42. Electrophorus 5
43. Gold leaf electroscope 5
44. Glass rod 1
45. Ebonite rod 1
46. Silk and Cat Skin pieces 1
47. Proof plane 2
48. Slide wire Bridge 5
49. Potentiometer 5
50. One Way and two way keys 5
51. Resistance box 5
52. Rheoslat 5
53. Resistance Coil 5
54. Stading Key (two keys) 5
55. Tangent Galvanometer 7
56. Induction coil 5
57. Torch lamp holder 2
58. Switches 1
59. cutout fuses 1
60. Pliers 5
61. Spirit Lamp 4
62. Tripod Stand 5
63. Retort stand and clamps 5
64. Tunning Forks 2
65. Resonance apparatus 5
66. Stove (Oil) 5
67. Binoculars 10
68. Soldering Rod (Fire) 5
69. Solder (Electric) 2
70. Graduate cylinder 2
71. Glass Plate Machine 5
72. Spirit Level 5
73. Battery Clamps 3
74. Siren 7
75. Hydrometer 5
76. Lactometer 5
77. Drawing Board 2
78. Barometer Tube 2
79. Photographic Camera 20
80. Telescope 20
81. Newton’s Disc 10
82. Pin hole camera 10
83. Microscope 30
84. Epidiascope 20
85. Radio set 10
86. TV. Set 10
� University Physic : H.D. Young, M.W.Zemansky and F.W. Sears, Narosa Pub. House.
� Physics - Foundations and Frontiers : GeorgeGamow and J.M. Ciearland, Tata Mcgraw Hill.
172
No.
& N
ame
of
the
Art
icle
Tota
l L
ife
in l
ived
life
of t
heA
rtic
les
Fixe
dV
ide
lette
rN
o.ac
cord
ance
of 4
(52)
61
Edn
Dat
ed2.
8.82
S. N
o &
Pag
e in
the
Prop
erty
Reg
iste
r
Whe
ther
the
artic
les
men
tione
d in
Col
. 9
have
com
plet
edth
eir
life
inac
cord
ance
with
the
life
men
tione
din
Co.
5S.
No.
Page
Dat
e of
purc
hase
Nor
mal
inbu
ildin
g
used
in
tent
s/do
uble
Shif
ts
Nam
eN
o.N
o. Tota
l
requ
ired
in s
tock
requ
ired
to
be w
ritte
n of
f
No.
/qua
ntity
artic
le i
nSt
ock
Bal
ance
of
serv
icea
ble
Rat
e as
per
stoc
k/pr
oper
tyR
egis
ter
Tota
l co
st o
f th
ear
ticle
s re
quir
edto
be
wri
tten
off
Rem
arks
if
any
affe
ctin
g th
e lif
eof
the
art
icle
sno
t co
vere
d by
othe
r C
olum
ns
Rec
omm
enda
tion
of t
he h
ead
of t
heIn
sitit
utio
nPa
ge
12a
2b3
4a4b
56
7a7b
89
1011
1213
14
PR
OF
OR
MA
OF
CO
ND
EN
MA
TIO
N
Nam
e of
the
Sch
ool—
——
——
—
Lis
t A
(W
hose
Lif
e H
as b
een
Lai
ddow
n) (
Uns
ervi
ceab
le A
rtic
les)
Lis
t of
Non
-con
sum
able
Con
sum
able
/Non
-Con
sum
able
Art
icle
s To
be
Wri
tten
Off
173
Cer
tifi
cate
:
1.C
ertif
ied
that
the
art
icle
s m
entio
ned
in t
he C
ol 9
hav
e be
com
e un
serv
icea
ble
due
to n
orm
al w
ear
and
tear
cer
tific
ate
furt
her
that
----
----
the
re h
as n
ot b
een
any
negl
igen
ce l
eadi
ng t
o br
eaka
ge o
f or
dam
age
to t
he a
rtic
les
on t
he p
art
of a
ny o
ffic
er c
once
rned
with
the
cha
rge
of t
he a
rtic
les,
2.C
ertif
ied
that
the
no.
san
ctio
n in
res
pect
of
the
artic
les
in q
uest
ion
was
rec
eive
d by
the
sch
ool
conc
erne
d in
the
par
t w
as a
nyof
the
se a
rtic
les
incl
udes
in
list
pend
ing
sanc
tion.
3.C
ertif
ied
that
in
the
abov
e st
atem
ent
only
tho
se i
tem
s ha
ve b
een
incl
uded
who
se l
ife
has
been
fix
ed v
ide
lette
r N
o. F
4 (5
2)76
1E
dn 2
-8-6
2 as
am
ende
d.
4.C
ertif
ied
that
the
art
icle
s m
entio
ned
in C
ol.
9 ha
s al
read
y co
mpl
eted
the
ir l
ife
as f
ixed
vid
e le
tter
No.
F(5
2)/6
1 E
dn.
2-8-
62am
ende
d.
5.It
em N
os.
----
----
---
have
bee
n ph
ysic
ally
ver
ifie
d to
be
unse
rvic
eabl
e an
d ar
e re
com
men
ded
for
bein
g w
ritte
n of
f.
6.C
ertif
ied
that
the
art
icle
s co
ntai
ned
in t
he l
ist
have
bee
n ch
ecke
d fr
om t
he s
tock
reg
iste
r an
d th
e da
te o
f th
e pu
rcha
se p
rice
etc
.ha
ve b
een
cert
ifie
d.
Mem
ber
of t
he C
onde
mna
tion
Boa
rd
174
Suggseted Readings
● Factes of Physics : A conceptual Approach, West Publishing Company.
● Fundamentals of Physics : David Halliday, Robert Resnick, Jearl Walker, Asian Books Pvt. Ltd.,New Delhi.
● University Physics : H.D. Young, M.W. Zemansky and F.W. Sears, Narosa Pub. House.
● Physics - Foundations and Frontiers : George Gamow and J.M. Clearland, Tata Mcgraw Hill.
● College Physics : R L Weber, K.V Manning, M.W. White & G.A Weygard, Tata Mcgraw Hill.
● College Physics : Reymond A. Sarvey and Jerry S. Fanghan Harcourt Brace & Co. Principlesof Physics : Raymond A. Serway & John W. Jewett, Jr.
● The Elements of Physics : I.S Grant & W R. Phillips.
● Physics can be fun : Y. Perelman, Mir Publishers.
● Advanced level Physics M. Nelkon & RParker, Arnold - Heinemann.
● Success in Physics - Tom Duncan, John Murray Publications Ltd.
● Success in Electronics - Tom Duncan : John Murray Publications Ltd.
● Concepts of Physics - H.C. Verma : Bharti Bharti Bhawan Publishers.
● 3000 Solved Problems in Physics : Alvin Halpern.
● Schaum’s solved problem series : Tata Mcgraw Hill.
● Mcgrwaw Hill’s Dictionary of Physics.
● Physics - Resnick and Halliday
Net Resources
The following is just a suggestive list:
● www.physicsclassroom.com
● www.learn4good.com/kinds/high_school_science_physics
● www.hsphys.com
● faraday.physicssuiowa.edu/resource.html
● www.library.aucktand.ac.nz/subjects/physics/phymeta.com
● http://en.wikipedia.org/wiki/Optics
● http://www.ee.umd.edu/~taylor/optics.htm
175
● http://www.walter-fendt.de/ph14e/mwave.htm
● http://www.school-for-champions.com/science/ac.htm
● http://www.aiiaboutcircuits.com/vol_2/cht_1/1.html
● ed.org/Education Resources/HighSchooi/Electricity/altematingcurrent.htm
● http://theory.uwinnJnpeg.ca/physics/bohr/node1.html
● http://en.wikipedia.org/wiki/Atomic_physics
● http://en.wikipedia.org/wiki/Wage-particle_duaiity
● L11/L8/de_Broglie_Wages/de_brog!ie-waves.htm!
● http://physics.nrnt.edu/~raymond/ciasses/ph.13xbook/node189.html
● http://www.search.com/refernce/Geiger%2DMarsden_experiment?redir=1
● http://einstein.stanford.edu/content/faqs/maser.html
● http://en.widipedia.org/wiki/Electromagneticjadiation
● http://www.eo.ucar.edu/rainbows
● http://www.atoptics.co.uk/bows.htm
● http://astro.nineplanets.org/bigeves.html
● http://users.ece.gatech.edu/~alan/ECE3080/Lectures/ECE3080-L-1-Introduction%20to%20Electronic%20Materials%20Pierret%20Chap%201%20and%202.pdf-GeorgiaTech
● www.pptsearch365.com/diode-properties-ppt.html
● ebookfreetoday.com/bulk-semiconductor~0~ppt.html
● downppt.com/ppt/semiconductor.html
and many more which you may get on the internet