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Zero syllabus

ifjp;&ifjp;&ifjp;&ifjp;& dksbZ Hkh fo"k; i<+uk ;k i<+kuk vkjaHk djus ds iwoZ gesa mldh i`"BHkwfe ds laKku dk gksuk

vR;ar vko';d gSA ;g laKku fo'ks"kdj foKku laca/kh fo"k;ksa ds fy;s T;knk egRoiw.kZ gS D;ksafd bu fo"k;ksa dh ikfjHkkf"kd 'kCnkoyh vU; lkekftd fo"k;ksa dh 'kCnkoyh ls dbZ ekeyksa esa fHkUu gksrh gSA

;g crkus dh vko';drk ugha fd fdlh Hkh fo"k; dks le>us ;k mls izLrqr djus ds fy;s rduhdh vkSj ikfjHkkf"kd 'kCnksa ls ifjfpr gq, cxSj vkxs c<+uk vkSj fo"k; esa izokg dks cuk;s j[kuk dfBu gksrk gSA ifjfpr 'kCnksa ds lkFk tc v/;;u@v/;kiu dh izfdz;k vkjaHk gksrh gS rc fo"k;&izos'k vklku gks tkrk gS o fo"k; esa Lor% ,d izokg cu tkrk gSA ;g ifjfpr jkLrs ij pyus tSlk gksrk gSA

tc le> ds Lrj ij lQyrk feyus yxrh gS rc f'k{kd@fo|kFkhZ vkuUn ls jksekafpr gksus yxrs gSaA lp rks ;g gS fd fo"k; ds v/;;u@v/;kiu ds nkSjku ifjfpr 'kCnksa ds feyus ls cksfj;r ugha gksrh] /;ku vklkuh ls cuk jgrk gS vkSj eu ds HkVdus ij Lor% jksd yx tkrh gSA

lkekU;r% foKku laca/kh fo"k;ksa esa HkkSfrdh ls fo|kFkhZ dqN vf/kd gh Hk;Hkhr jgrs gSaA tcfd ckr ,slh gksuk ugha pkfg;sA HkkSfrdh rdZ ij vk/kkfjr fo"k; gS ftlds v/;;u esa mu ?kVukvksa dk lekos'k gksrk gS ftUgsa ge vius vklikl ?kVrs gq, ns[krs gSa] lqurs gSa vkSj eglwl djrs gSaA HkkSfrdh; v/;;udrkZ dk mn~ns'; ftKklk iSnk djus okyh gj ?kVuk dks le>uk vkSj muds ihNs fNis dkj.kksa dks tkuuk gksrk gSA ,slk gksus ls izd`fr ds fu;eksa dk irk pyrk gSA HkkSfrdh; v/;;udrkZ fQj mu fu;eksa dks vk/kkj eku dj vuqiz;ksx djuk pkgrk gS vkSj vkxs fNih ubZ laHkkoukvksa dks ryk'kuk pkgrk gSA og viokn Lo:i mu ?kVukvksa dks Hkh tkuuk pkgrk gS tgk¡ ;s fu;e dke u djrs gksaA

bl rjg vc vxj ewy :i esa ns[ksa rks gesa vius vklikl inkFkZ] ÅtkZ o vkdk'k fn[krs gSa vkSj dky cgrs gq, izrhr gksrk gSA vr% HkkSfrdh gesa inkFkZ vkSj mldh xfr ls tqM+k foKku utj vkrk gSA inkFkZ pkgs vkosf'kr gks ;k fujkosf'kr] lw{e gks ;k LFkwy & HkkSfrdh; n`f"V mls fof'k"V cuk nsrh gSA blhrjg xfr;k¡ pkgs og LFkkukUrj.kh; gks ;k ?kw.khZ ;k nkSyuh;] HkkSfrdh; n`f"V iM+us ij os izd`fr ds jkt [kksyrs utj vkrh gSaA tSls tSls gekjs lkeus izd`fr ds jkt [kqyus yxrs gSa gesa vikj vkuUn dh vuqHkwfr gksus yxrh gS ftlls HkkSfrdh esa gekjh :fp c<+us yxrh gSA ysfdu ;g fugk;r O;fDrxr ekeyk gSA vkjafHkd dfBukb;k¡ rks vkrh gSa ysfdu bUgsa ikj djrs gh etk vkus yxrk gSA

HkkSfrdh; v/;;u esa gj ?kVuk ij /;ku nsuk] ladsrksa dks xzg.k djuk] fo'ys"k.k djuk vkSj varlZEca/kksa dks [kkst dj fdlh rdZlEer fu"d"kZ ij igq¡puk gksrk gSA blrjg HkkSfrdh; v/;;u inkFkZ dh cká nqfu;k ls inkFkZ dh vkarfjd nqfu;k esa izos'k dk jkLrk lqyHk djkrk gSA ;g v/;;u i<+us&i<+kus okyksa dks ogk¡ ys tkuk pkgrk gS tgk¡ og ns[kk tk lds tks vk[kksa dh {kerk ls Hkh ijs gks] tgk¡ og lquk tk lds tks dkuksa dh {kerk ls Hkh ijs gks vkSj tgk¡ og eglwl fd;k tk lds tks dkWeu lsal ls Hkh ijs gksA blds fy;s HkkSfrdfon~ ekWMy dk lgkjk ysrs gSa] fl)kar x<+rs gSa vkSj ifjdYiuk cukrs gSaA fQj os mUgsa tk¡prs&ij[krs gSa vkSj HkkSfrdh dk <+kpk [kM+k djrs gSaA

gj d{kk dk viuk ikB~;dze gksrk gSA mldk vk/kkj iwoZ dh d{kkvksa esa i<+s x;s ikB~;dze gksrs gSaA fcuk vH;kl ds cht&'kCnksa vkSj ikfjHkkf"kd o rduhdh&'kCnksa ds vFkZ ;kn ugha jgrsA ;g i<+us&i<+kus esa ck/kk mRiUu djrk gSA blhfy;s vko';d gS fd ge tc bUgsa le>uk vkjaHk djrs gSa rks LokHkkfod gh dqN dfBukb;k¡ lkeus vkus yxrh gSaA blls cpus ds fy;s ;g vko';d gS fd i<+us&i<+kus ds iwoZ ge mu cht&'kCnksa vkSj rduhdh&'kCnksa dks tku ysa ftuls iwjs ifjp; ds fcuk

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gekjk fo"k;&izokg lgt ugha jgrk gSA vkSj] fcuk lgt izokg ds u gh eu yx ldrk gS vkSj u gh i<+us&i<+kus esa vkuUn fey ldrk gSA

HkkSfrdh ds fy;s 'kwU; ikB~;Øe dh jpuk&HkkSfrdh ds fy;s 'kwU; ikB~;Øe dh jpuk&HkkSfrdh ds fy;s 'kwU; ikB~;Øe dh jpuk&HkkSfrdh ds fy;s 'kwU; ikB~;Øe dh jpuk& HkkSfrdh ds fy;s 'kwU; ikB~;Øe dh jpuk djus ds ihNs gekjh blh Hkkouk us dk;Z fd;k gSA

blesa mu cht&'kCnksa vkSj ikfjHkkf"kd o rduhdh&'kCnksa ds lkFk gh xf.krh; rduhdksa dks lfEefyr fd;k x;k gS ftudh t:jr d{kk 11oha ds HkkSfrdh ds ikB~;dze dks i<+rs le; gksrh gSA

;g 'kwU; ikB~;dze yphyk gSA blesa cgqr dqN tksM+k vFkok ?kVk;k tk ldrk gSA ;g iwjh rjg vki ij fuHkZj gSA vxj i<+krs le; vki eglwl djrs gSa fd cPps HkkSfrd jkf'k;ksa dh foek vkSj bdkbZ le>us esa Hkzfer gksrs gSa ;k dfBukbZ eglwl djrs gSa rks 'kwU; ikB~;dze esa bUgsa foLrkfjr :i esa 'kkfey fd;k tk ldrk gSA ysfdu vxj os xf.krh; rduhdksa ds vuqiz;ksx ls ?kcjkrs gSa rks 'kwU; ikB~;dze esa bu ij fo'ks"k /;ku fn;k tk ldrk gSA

'kwU; ikB~;dze dh le;kof/k fuf'pr ugha dh tk ldrh gSA ;g rc rd pyk;k tkuk pkfg;s tc rd fd cPpksa ds dkUlsIV~l vPNs u gks tk;saA fcuk vPNs dkUlsIV~l ds HkkSfrdh i<+us dk mn~ns'; iwjk ugha gks ldrk gSA ;g f'k{kd ij fuHkZj djrk gS fd mls fdl fgLls ij fdruk /;ku nsuk gSA bl ikB~;dze dk eq[; mn~ns'; gS fd fo|kFkhZ i<+rs le; fo"k; esa lgt izokg eglwl djsa vkSj vius esa :fp iSnk djrs gq, ftKklq curs pys tk;saA

General terms

Axiom ¼Lo;afl) dFku½¼Lo;afl) dFku½¼Lo;afl) dFku½¼Lo;afl) dFku½: it is a mathematical statement which is assumed to be truth without proof falsify is yet to be established Conjecture¼vuqeku yxkuk½¼vuqeku yxkuk½¼vuqeku yxkuk½¼vuqeku yxkuk½: it is a mathematical statement whose truth or falsify is yet to be established Hypothesis ¼ifjdYiuk½¼ifjdYiuk½¼ifjdYiuk½¼ifjdYiuk½: proposition made as a basis for reasoning without the assumption of its truth. It is supposition made as a starting point for further investigation from known facts. It may be called groundless assumption. Law ¼fu;e½¼fu;e½¼fu;e½¼fu;e½: a statement of fact to the effect that a particular phenomenon always occurs if certain conditions are present Proof ¼izek.k½¼izek.k½¼izek.k½¼izek.k½: A process which can establish the truth of a statement based purely on logical arguments Theorem ¼izes;½¼izes;½¼izes;½¼izes;½: It is a mathematical statement whose truth has been logically established

Meaning of the words that appear in questions

Discuss: examine and argue about Describe: give a picture in words Elaborate: describe in detail with much care Write: put down by means of words Generalization: more common or more widely applicable Linearization: Able to be presented by a straight line on a graph

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What is physics? Its origin is latin word ‘physica’. Its meaning is ‘natural things’. Physics deals with

matter, energy, space and time. System Subsystem Environment

Physical Parameters

Parameters related to translatory and rotatory motions

Adhesion: its meaning is sticking to surface. The adhesive force is the inter molecular attraction between unlike molecules.

Atomic mass: sum of the neutron and protons Atomic number: number of protons

Capacity: the inner volume of a container

Central force: r-dependent forces are known as central forces. The gravitational, electrostatic and magnetic forces are examples of central forces.

Centre of mass: the point at which the mass of the body may be considered to be concentrated. It is the point from which the sum of the moment of mass (product of the mass of element and its distance from the centre of mass) of all the elements of the body is zero.

Cohesion: its meaning is holding together. The cohesive force holds a solid or liquid together due to inter molecular attraction. This force decreases with rise in temperature.

Degrees of freedom: the number of independent ways in which a system may possess translational, vibrational and rotational motions.

Density: In general, it is ‘mass density’ called as only ‘density’ and is defined as mass per unit volume. But for uniform surface, it may be surface mass density and is defined as mass per unit surface area. For a uniformly thin rod, it may be ‘linear mass density and is defined as mass per unit length. It may be defined to express the density of charge. For charge it is called ‘volume charge density’ and is defined as ‘charge per unit volume. The surface and linear charge densities are defined accordingly.

Different kind of forces: in general, we have two types of forces: contact and non-contact forces. The familiar forces are gravitational, mechanical, frictional, cohesive, adhesive, muscular, nuclear, electrostatic, magnetostatic, electromagnetic, nuclear etc.

Distance and displacement: the distance is scalar quantity but the displacement is vector. If a particle moving on a circular path completes one round, then the displacement of the particle is zero but the distance covered by it is 2πr

Elastic collision: it is the collision in which conservation of momentum and kinetic energy hold. In other words, for such collision, momentum and kinetic energy before collision are equal to the momentum and kinetic energy after collision. During inelastic collision, kinetic energy is not conserved.

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Elasticity: the property of the body of resuming its original form and dimensions when the impressed forces upon it are removed. If the forces are sufficiently large for the deformation to cause break in the molecular structure, it loses its elasticity and the elastic limit is said to be reached. Hook’s law (stress is proportional to strain) acts only within elastic limit. The behaviour opposite to that of elasticity is known as plasticity. No any material is 100% elastic or plastic.

Energy: ability to do is called energy. Its unit is joule. It may have different forms. It may be light energy, or sound energy, electromagnetic energy, heat energy, magnetic energy, electrical energy, chemical energy, nuclear energy. In general the energy is classified as kinetic energy and potential energy. If we do some work on the body, it is stored as potential energy in the body. Potential energy is a measure of capacity of the body to do the work. The kinetic energy is possessed by the moving bodies. For translational motion it is equal to ½ mv2 and for rotational motion it is ½ Iω2.

Equations of motion: v = u + at; s = ut +1/2 at2; v2 = u2 + 2as.

Equilibrium: A state of balance between opposing forces or effects

Field: the region in which a massive body, electric charge, a magnetic body exerts its influence. A field is actually a model for representing the way in which a force can exist between bodies not in contact.

Floatation, principle: (Archimedes principle) the weight of the liquid displaced by a floating body is equal to the weight of the body. If the weight of the floating body is more than the weight of the liquid displaced, the body sinks.

Fluids: Gases and liquids together are called fluids.

Force: It is a vector quantity. It is defined using Newton’s second law of motion. According to this, the rate of change of momentum is called force. Or, it is the differential coefficient of momentum with respect to time. Its unit is Newton.

Friction due to liquids and gases: Liquids exerts less force of friction and the gases the least as compared to solid.

Friction: An opposing force that appears when two surfaces in contact with each other try to move relative to each other. Friction is a necessary evil. Without friction neither the bodies can be set in motion nor can the moving bodies be stopped. Use of lubricants, soap solutions, fine powders and polishing can reduce friction.

Friction: It is the force that offers resistance to relative motion between surfaces of contact. The static coefficient of friction is the ratio of the frictional resistance when the body is about to slide along a specified surface (Fs) to the perpendicular force (R) between the surface in contact. The kinetic coefficient of friction is the ratio of the frictional resistance when the body is sliding steadily along a specified surface (Fk) to R.

Fundamental forces of nature: Gravitational, electromagnetic, nuclear strong and nuclear weak forces.

Graphs and extrapolation of the curve in the graph

Impulse: It is the product of force and the duration during which it acts.

Isotope: The elements having same atomic number but different atomic mass numbers.

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Latitude: The angular distance of a point from the equator measured upon the curved surface of the earth. The latitude of equator is zero and that of poles are 90o.

Law of conservation of energy: Energy before event is equal to energy after event. This law is a direct consequence of third law of motion.

Law of conservation of momentum: Momentum before event is equal to momentum after event. This law is a direct consequence of third law of motion.

Laws of motion: There are three laws. First law: if a body continues to remain in the state of rest or uniform motion unless and until an external force acts on it; Second law: the rate of change of momentum is force; Third law: to every action there is an equal and opposite reaction.

Lever: It is a bar that is free to move about the point (called fulcrum) on which it rests. The object to be moved is called load or resistance and the force applied is called effort. There are three classes of lever.

Linear: Means arranged in a line. It has only one dimension. The system is known to be linear if the output is proportional to input.

Longitude: The angle which the terrestrial meridian through the geographical poles and the point on the earth’s surface makes with a standard meridian (passing through Greenwich). Machine: It is a devise which makes work easier to do. There are six simple kinds of machines viz. lever, inclined plane, wedge, screw, wheel-and-axel and pulley

Manometer: It is a devise to measure the pressure.

Meteors: Shooting stars

Moment of force: It is called torque. It is a measure of the tendency of a force to rotate the body to which it is supplied. It is measured by multiplying the magnitude of the force by the perpendicular distance from the line of action of the force to the axis of rotation.

Moment of inertia: It is the sum of the products of the mass dm of each element of the body (a body is assumed to be made of a large number of infinitesimal elements of mass dm) and square of its distance from the axis of rotation.

Moment of momentum: It is angular momentum and is equal to product of moment of inertia and angular velocity

Momentum: It is a vector quantity. it is the product of mass and velocity.

Power: It is rate of doing work. Its unit is joule/second. it is also known as watt.

Pressure: It is defined as force per unit area. Its unit is Newton/m2 and is called ‘pascal’. The pressure increases with depth in the liquids. Deep sea divers have to wear specially designed suits, otherwise the huge pressure of water will crush their bodies. The dams are thickened at the base to withstand huge pressure. At high altitudes, air pressure is very small.

Pull and Push: These are forces. Push or pull changes the speed and direction. In other words, they change velocity.

Pulse: A brief increase in the magnitude of a quantity whose value is usually constant

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Quantities that characterise motion: Displacement, velocity, momentum, acceleration, force, kinetic energy and potential energy

Relative density: It is the ratio of density of substance to the density of water.

Rolling friction: It is the friction experienced by the body when it is made to move over bodies like roller. It is far less than sliding or dynamic friction (about 10 times). The ball bearings (small steel balls) are introduced between the sliding surface of the axel and the cup.

Speed and velocity: Speed is scalar quantity but velocity is vector. The speed is rate of change of distance covered but the velocity is the rate of change of displacement

Speed: It is the rate of change of distance covered. Speed = distance/time. It is also defined as differential coefficient of distance with respect to time.

States of matter: In general, there exists four states of matter, viz., solid, liquid, gas and plasma. This classification is based on general properties of matter. However, matter may be classified on the basis of other properties. On the basis of optical properties the states are: opaque, translucent and transparent. On the basis of electrical properties the states are: insulator, semiconductor, conductor and superconductor. The classification can also be made on the basis of thermal and magnetic properties.

Stethoscope: It is a devise that amplifies the sound of a heart beat and other sounds in the chest and back.

Stream line and turbulent flow: The type of fluid flow in which the motion at any point varies in magnitude and direction. The fluid flow in which this doesn’t happen is called stream line flow.

Thrust: The compressive force applied by the material. Uniform motion: When the speed remains constant, we call motion to be uniform. Non-uniform motion results if the speed is not uniform.

Universal law of gravitation: 2

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21

r

mmGF =

Velocity: It is the rate of change of displacement. Or, it is differential coefficient of displacement with respect to time.

Viscous: Having high viscosity. Viscosity is the property of fluid whereby it tends to resist relative motion within itself. The fluid may be modelled as made of thin layers. If different layers of fluid are moving with different velocities, viscous forces come into play, tending to slow down the faster moving layer and to increase the velocity of slower moving layers. The viscous force is proportional to the velocity gradient between the layers. The constant of proportionality is called ‘coefficient of viscosity’. Weight of the body: F = mg Work: It is said to be done only if the force moves the object. It is a scalar quantity and is defined as a scalar or dot product of force and displacement.

Parameters related to waves and oscillations

Amplitude: It is the maximum displacement of vibrating body.

Audible range of sound; 20 to 20000 herz.

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Coupling: Connecting the two systems; coupled oscillators are two oscillators coupled by , say, a spring.

Damped and undamped vibrations: Vibrations whose amplitude doesn’t change with time are known as undamped vibrations. But if amplitude changes with time, they are damped vibrations.

Damping: A decrease in the amplitude of an oscillation or wave motion with time.

Doppler effect: The apparent change in frequency due to relative motion between source or listener.

Echo: The reflection of sound from hills or big buildings is called echo.

Frequency; it is the number of oscillations per second. n = 1/T

Infrasound: The sound whose frequency is less than 20 hertz. It is not audible to human ears

Intensity of sound: It depends upon amplitude. It also depends upon the inverse square distance from the source and area of the vibrating body. I ∝A2 , I ∝1/d2 and I ∝ S. the loudness depends upon the density of medium also. Lesser the density, lesser loud is the sound.

Longitudinal: Lengthwise. In a line with the length of the object. Longitudinal vibration of a body is that in which the line of oscillation coincides with the length of the body

Loudness and pitch: Frequency of vibration determines the shrillness or pitch of the sound. Loudness of sound depends on the intensity of sound and sensitivity of ear. The intensity of sound and hence loudness is measured in decibels (dB). Human ears can pick up sound from 10 -180dB. The loudness is considered to be normal within the range 50 -60dB. Humans can tolerate up to 80dB. Beyond this range, it is painful and so considered as undesirable. The sound with loudness greater than 80dB creates noise pollution. The noise pollution is determined not only by loudness but also by its duration. I ∝ L. Musical sound: a sound that has a pleasing effect on the ears Natural frequency: The frequency of free oscillation of system

Noise: A sound that has an unpleasing effect on the ears

Periodic and non-periodic changes: To and fro motion of a ball is called oscillation. If the changes are repeated after a fixed time interval, the changes are called periodic. The changes like formation of hurricane or occurrence of earth quake may occur any time. There occurs no periodicity and hence they are known as non-periodic.

Reflection of sound: Sound gets reflected from the smooth surfaces. It follows the law of reflection i.e. angle of incidence is equal to angle of reflection.

Resonance: When the natural frequency coincides with the external force frequency, we get resonance.

SONAR: Sound Navigation and Ranging

Superposition of waves: y = y1 +y2

Time period: It is the time required to complete one oscillation

Transverse: Crosswise; in a direction at right angles to the length of the object.

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Transverse vibration of a body is that in which the line of oscillation makes an angle of 90o with the length of the body

Ultrasound: The sound whose frequency is greater than 200000 hertz. .It is not audible to human ears. They are used as diagnostic tools.

Velocity of wave: λλn

Tv ==

Wave length: vT=λ

Parameters related to heat and temperature

Absorption of heat: It depends on the mass of body, change of temperature and nature of substance (expressed in terms of specific heat ‘s’). )( TmsQ ∆=

Adiabatic process: It is the process in which no heat is exchanged between the system and the surrounding environment

Bi-metallic strip: It is made of equal lengths of steel and copper strips.

Boiling is a bulk phenomenon but evaporation is surface phenomenon but both require latent heat. In both the processes liquid gets converted into gaseous state. Interestingly, boiling takes place at fixed temperature but evaporation takes place at all temperatures below boiling point. Rate of vaporization of water increases with temperature, decreases with increase in humidity and increases with increase in wind speed.

Calorie and joule: Calorie is the amount of heat required to raise the temperature of one kg of water by 1oC.

Clinical thermometer: It is used to measure body temperature. Its range varies from 35 to 42oC.

Effect of heat: Effects of heat flow are rise in temperature, change in size, change in volume (water expands after 4oC but it contracts when heated from 0oc up to 4oC), increase in pressure, change of state, chemical change, physiological change etc.

Efficiency: It is the ratio of output to the input

Gas (ideal) equation: p v = RT Heat and temperature: Heat is a form of energy but temperature is the degree of hotness.

Heat capacity: It is the amount of heat required to raise the temperature of the body by 1oC. heat capacity = m s.

Isobaric process: The thermodynamic process taking place at constant pressure.

Isochoric: The thermodynamic process taking place at constant volume.

Isothermal : The thermodynamic process taking place at constant temperature.

Latent heat: It is associated with the change of state. The heat given to the system does not increase its temperature.

Liquefaction point: It is the temperature at which gaseous state starts converting into liquid state without any change in temperature.

Modes of heat transmission: Conduction, convection and radiation

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Recognition of parameters that show temperature dependence: thermal expansion (Mercury and Alcohol thermometers), pressure (Air thermometer), electrical resistance (Platinum resistance thermometer), thermo e.m.f. (Thermocouple, thermopile) etc.

Solidification point: It is the temperature at which liquid starts converting into solid state without any change in temperature.

Specific heat: It is the amount of heat required to raise the temperature of unit mass of substance through a unit temperature.

Temperature (relation between Celsius and Fahrenheit) 9

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−= FC

Thermal expansion: The expansion produced in the substance when heated is known as thermal expansion.

Thermostat: It is a devise used to control the temperature automatically.

Green house: It is used to grow plants in cold weather.

Parameters need quantitative expression

Standards Measurement Errors Units (C.G.S. and M.K.S. system)

Awareness of mathematical tools

Graphs: Choosing two variables keeping rest constant; x,t; p,v; v,t etc. Area under the curve Geometry Algebra From geometry to algebra Logarithm Trigonometry Integral and differential calculus Vectors Difference between addition of vectors and numbers Shape and symmetry Groups Criterion of forming groups Parameters need of measurement directly or indirectly

Errors in measurement Choosing suitable instrument Least count Quality and Quantity Dimensions One, two and three dimensions: line, plane and solid body Magnitude and direction

Studies in Physics

Understanding of an event qualitatively

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Knowing the variables and the environment Inputs and outputs Recognition of the scalar and vector nature of the involved quantities

Dealing with the parameters Finding interrelations Description of the event Choosing a coordinate systems Formulation of the problem Proceed to solve

Approximations

Linearity is a good approximation but it fails at number of occasions Newton’s laws fail at high speeds Mass is speed independent – good approximation in day to day life motion

Leaning to Model

Knowing the symmetries of the system Ring, disc, shell, sphere, cylinder, Rigid body Black body Ideal and real gas Atom, molecule Monatomic, diatomic and polyatomic Interatomic and intermolecular distances and forces

Problem phobia

How to get rid of Problem phobia?

The student's possess a poor level of problem solving. They are afraid of problems. This is not a good state of affair with the growth of physics education. Merely exposure to the large number of topics is not going to help in making physics as career. Unless and until one has extensive knowledge, he can not be successful. The problem solving skills help in having depth of understanding. The reason being the examination system. Even the autonomous colleges get their paper set from those who are involved in the conventional routine type of examination system. The university examination system is designed to get easy grading. The students can easily guess what is to be expected from them in the examination simply by looking into last two - three years' question papers. The aim of the students is to pass with good marks with or without depth. For them, cramming and memorizing are only important and not the critical thinking and creativity. The heaviness of the course contents encourages them to follow this easy path. For teachers also it is easy to teach with examination point of view. When they teach in examination friendly style, there remains no scope for creativity in the class.

In the autonomous colleges, there is flexibility in designing the course contents. It may be remembered that the course contents should provide a general guideline only and should not become Geeta for you because for creativity, flexibility is necessary and you need to go left and right depending upon the level of the class. This flexibility gives scope for the problems to be discussed in the class. This is only the way of teaching science creatively. Further, one should remember that discussion on problems does not mean to solve the problem mathematically step by step. In stead, the various possible approaches need to be discussed leading to thorough understanding of the subjects. The implications of various approaches are to be discussed. There should be flow during teaching but spoon feeding need to be avoided at any cost. This is only the way to bring self-reliance in the students. Simple calculations and obvious steps should intentionally be avoided and should be given as homework to the students. The time saved in this way can easily be invested in developing problem solving skills in the students.

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Further, the preparing, scoring and evaluating the examination must also be done by the same teacher or a board of teachers appointed by the department in which course instructor should be one of the members. This will provide a great deal of flexibility to the teacher in the classroom while teaching. Truly, the major concern of Physics teaching should be to expose the students with the art of probing, observing and analyzing. The examination should be given to test the degree of perfection achieved.

The purpose of teaching is to make aware the students of their hidden potential so that they can solve the real life problems that will always be new and in appear during unforeseen circumstances.

Study the problem

Read the given problem carefully. This will help in thorough understanding of the problem and in extracting what is given and what is to be determined. Following steps may be followed:

Find the area in which the problem belongs

Recall the background in your memory

Set various help levels

Split the problem into suitable parts

Recognize help levels for each part of the problem

Diagnose the problem

Make list of the givens. If graph is given, read it carefully and extract all possible information from it.

Convert all the units of the given quantities in any one convenient system of units and follow the same throughout.

Search for the hidden information: symmetry of the problem: choosing the coordinate system: polar or Cartesian or elliptical or parabolic. Try to get a geometrical picture, which can be used to find relations between various quantities.

What is to be determined which can lead to the solution of the problem?

Is there any direct relation between the known and unknowns to be determined?

Is there any scope for taking any approximation using Bionomial theorem, Taylor's theorem, logarithmic, exponential or sin -cosine series? Make a note. Bionomial theorem can be used to estimate approximate value the square root of any number (1 + x)1/2, where x <<1, as (1 + x/2).

Result and significant figure:

In general, in the final result one should not have more significant figures after decimal than in the original data. But the process of rounding off must be done only in the final result and not during the calculation without proper physical justification.

Technique to be followed:

1. Look into those direct problems, which require direct application of formula. This is necessary to be familiar with the types of calculations and proper selection of the units required.

2. Look into those direct problems, which require indirect application of formula. This is necessary to inculcate the habit of searching some hidden links.

3. Look into those direct problems, which require some approximations to be considered before applying the formula. This will help in knowing the importance of the terms with physical point view that can affect the results.

4. Start with the problems that require the given situation to be analyzed. Select those problems whose solutions are available but not known to you. Try to find the approach or approaches and attempt solving. Check the answer and compare the approach with that used in the available solution. This is necessary for adjudging the level of understanding you possess and gaining the confidence level.

12

5. After gaining confidence, take up unknown type of situational problems and then examine critically the approach.

Errors in measurement:

Some definitions

Instrument:

A device/mechanism used to measure the present value of a physical quantity

Measurement:

The process of determining the amount, degree or capacity by comparision with accepted standards of the system of units

Accuracy:

The degree of exactness of a measurement compared to the expected or desired value

Resolution:

The smallest change in the input to which an instrument responds

Sensitivity:

It is closely related to resolution. It is defined as the ratio of change in output of the instrument to the change of input of measured variable. An instrument may have very good sensitivity but very poor resolution or very small sensitivity but good resolution.

Precision (P):

It is a measure of repeatability or reproducibility. It gives consistency of instrument output for a given value of input. It is defined as

n

nn

X

XXP

−−= 1

Here nX is the average value of X and Xn is the value of nth observation.

Expected value:

The design value or most probable value (Yn)

Absolute error (e):

The deviation of the true or measured value (Xn) from the desired value (Yn)

e = Xn - Yn

Percentage error:

% error = 100×nY

e

Accuracy (A):

13

Error is also expressed as absolute accuracy

A = 1 - nY

e

This may be expressed as % accuracy.

Significant figures:

It is the indication of precision of measurement. Let the reading of temperature is 25oC. This indicates that the reading may vary from 24 to 26 oC. But, if the reading of temperature is 25.1oC. This indicates that the reading may vary from 24.9 to 25.1 oC.

Illustration:

1. The expected value of voltage across a resistor is 80V and the measured value gives 79V. Calculate the absolute error, percentage error, relative accuracy and percentage accuracy.

Answers: 1V, 1.25%, 0.9875, 98.75.

2. Let us consider following set of 10 observations and calculate the precision of the 6th measurement.

S.No. Measured value (Xn) 1. 98 2. 101 3. 97 4. 102 5. 101 6. 100 7. 103 8. 98 9. 99 10. 106

Average of 10 measurements = 1005/10 = 100.5

6th measurement: 100

Precision = 1- 5.100

5.100100−

Precision of the 6th measurement is 0.995

Errors

Static and dynamic errors

Static error of any measuring instrument is the numerical difference between true value and the experimentally observed value obtained by measurement.

Dynamic errors are related to the sluggishness of the instrument in giving response to the input signal. It is due to inertia, thermal capacitance, fluid capacitance and electrical capacitance.

14

This error is important while using industrial instruments. These instruments waits for certain process to take place. These instruments measure the quantities that always fluctuate in time. The dynamic behaviour is determined by subjecting its sensing element to some unknown and predetermined variations (e.g. stepwise changes, linear changes, sinusoidal changes etc.) in the physical quantity.

The dynamic characteristics of an instrument are

Speed of response Fidelity (faithful reproduction) Lag Dynamic error

Sources of error:

1. Insufficient knowledge of process parameters and design condition 2. Poor design 3. Poor maintenance 4. Human involvement in operation 5. Design limitations

Types of static error

This error may be classified as

Gross error

Systematic error

Random error

Gross error: Human mistakes (also carelessness and bad habits) in reading or using instrument (not setting zero) and in recording and calculating. Some gross errors may easily be detected but some are really illusive. These errors can not be treated mathematically. These can be avoided only through practice.

Systematic error:

These are generally due to shortcomings of instruments. Defective parts, worn parts, ageing effects of the environment on the instrument cause this type of error. This error is of three types;

Instrumental error Environmental error Observational error

The instrumental errors are due to the mechanical structure of the instruments. Friction in bearing, irregular spring tension, calibration errors etc. creates this type of error. The effects of changing environmental conditions e.g. temperature, pressure and humidity changes. The electrostatic and magnetic field’s present may also cause this error. These errors may also be divided as static (due to limitations of measuring device) and dynamic (due to slow responding) errors.

15

Random error: These have unknown origin. These are generally due to accumulation of a large number of small effects and may be of real concern if high degree of accuracy is required. They cannot be corrected by any method of calibration or any known method of control. It can not be explained without minute investigations. The only way is to collect a large number of data and then treating them statistically as these errors are due to unknown causes but follow the laws of probability.

Statistical treatment of data:

For these the systematic error in the data should be as small as possible as compared to random errors. The statistical treatment can not remove a fixed bias contained in all the measurements.

Arithmetic mean: The most probable value of a measured variable is the arithmetic mean. It is defined as

_ x1 +x2 +x3………xn x = ------------------------------- n

Deviation from mean: It is the departure of the given reading from the arithmetic mean. For example, the deviation of the first reading from mean, d1, is

_ d1 = x1 - x

Here a proper sign is used to express deviation. However the expression of the absolute value of deviation requires no sign.

If there are ten readings for an observation, there will be ten deviations (d1, d2, ….d10) from the arithmetic mean. By taking any suitable example, it can easily be demonstrated that the sum of all the deviation equals zero.

Average deviation: It is an indication of the precision of the instrument employed. It is defined as the average of absolute value of all the deviations for the readings of the observation Variance or mean square deviation: it is the mean of all squared deviations (d1

2, d22, ….d10

2) from arithmetic mean. It is expressed by the symbol σ2. Root mean square or standard deviation: it is the square root of the sum of all the individual deviations squared (d1

2, d22, ….d10

2) divided by total number of readings (n) for infinite number of readings. But if the readings are finite in number, then the sum of all the individual deviations squared (d1

2, d22, ….d10

2) is divided by (n-1) and not by ‘n’ in order to get standard deviation. It is expressed by the symbol σ.

Probability of errors: normal distribution of errors

The result of a series of measurements can be presented can be presented graphically in the form of a block diagram or ‘histogram’ in which the number of observations (frequencies of occurrence) are plotted against the readings of measured physical quantity. Say for example, we have following table for the 50 readings during current measurement in a circuit

Readings (in ampere) Frequency of occurrence 56.1 2 56.2 5

16

56.3 10 56.4 24 (most frequently occurred) 56.5 5 56.6 3 56.7 1

The largest frequency occurs at the central value corresponding to 56.4 ampere. The others are more or less placed symmetrically around this value.

If more readings are taken at smaller increment (say 0.05), the distribution would be observed to be more symmetrical (better than previous). A large number of data at still smaller increment will result into a very smooth bell-shaped curve in the plot. The contour of the histogram is perfectly symmetric. The bell-shaped curve is known as Gaussian curve.

The sharper and narrower Gaussian curve implies that the central value is most probable value of the true reading.

The Gaussian or normal law of error forms the basis of the analytical study of random effects.

According to this law: 1. All observations include random error (due to small perturbing effects of unknown

origin) 2. Random errors can be positive or negative 3. There is equal probability for positive and negative random errors.

The error distribution curve (Gaussian probability curve) is obtained by plotting error function (in terms of σ) against the occurrence of the readings. The curve based on normal law shows a symmetrical distribution of errors. It extends from -∞ to +∞. The area under the curve between + σ to – σ represents the cases that differ from the mean by no more than standard deviation.

Probable error: In the curve find the limits of error function that includes half the cases. This limit gives ‘probable error’. The probable error is used in experimental work in the past. However the standard deviation is preferred as it is more convenient in statistical work.

Limiting errors:

Look at the instruments. Manufacture gives a guarantee of accuracy. Similarly, look at electronic components. They are guaranteed within a certain percentage of their rated value.

The limits of the deviations marked on the instrument (as provided by manufacturer) are known as limiting errors (or guaranteed errors).

17

bdkbZ & 1 ekiu ,oa lfn'k&vfn'k

HkkSfrdh gS D;k\] izkphu Hkkjr esa ekiu] osx ladkj] fLFkfr LFkkidrk laLdkj] xq:Ro rFkk xq:Rokd"kZ.k] vfn’k rFkk lfn’k] lfn’kksa dk fo;kstu] f}in izes;] ekiu esa =qfV;k¡A

izLrkoukizLrkoukizLrkoukizLrkouk

foKku dk 'kkfCnd vFkZ gS fof'k"V KkuA foKku 'kCn dh O;qRifÙk ysfVu Hkk"kk ds

'kCn **Scientia** ls gqbZ ftldk vFkZ gS **tkuuk** Kku dk laca/k tkuus ls gksrk gSA

laLd`r 'kCn **foKku** vkSj vjch 'kCn **bYe** lekukFkhZ 'kCn gSA ftudk vFkZ gksrk gS

**laxfBr Kku**A

izd`fr vkSj izkd`frd ?kVukvksa ds Øec) izs{k.kksa ,oa iz;ksxksa ls izkIr Kku dks

foKku dgrs gSA rdZ vkSj iz;ksx foKku ds egRoiw.kZ vkStkj gSA

foKku dh 'kk[kk,¡& foKku dh 'kk[kk,¡& foKku dh 'kk[kk,¡& foKku dh 'kk[kk,¡& vk/kqfud ;qx esa foKku ds rsth ls fodkl o v/;;u dh lqfo/kk ds

fy, bls eq[;r;k nks 'kk[kkvksa esa foHkkftr fd;k x;k gS&

� HkkSfrdh; foKku ¼HkkSfrdh; foKku ¼HkkSfrdh; foKku ¼HkkSfrdh; foKku ¼Physical Science½& ½& ½& ½& blds vUrxZr HkkSfrdh] jlk;u foKku]

[kxksy foKku] vkfn fo"k;ksa dk v/;;u fd;k tkrk gSA

� thofoKku ¼thofoKku ¼thofoKku ¼thofoKku ¼Biological Science½& ½& ½& ½& blds vUrxZr thofoKku] ouLifr foKku

vkfn fo"k;ksa dk v/;;u fd;k tkrk gSA

;gk¡ ij ge dsoy HkkSfrd 'kkL= ds ckjs esa ppkZ djsaxsA

1111---- HkkSfrdh gS D;k HkkSfrdh gS D;k HkkSfrdh gS D;k HkkSfrdh gS D;k \\\\

jkf= dk vkdk’k ns[kks & jks’kuh esa ugk jgs 'kgj esa ugha xk¡o esa jg dj ns[kksA

izd`fr dh xksn esa cSB dj ns[kksA unh] igkM+ vkSj lkxj dh vksj ns[kksA izd`fr esa ?kV

jgh ?kVukvksa dks ns[kksA euksgkjh n’;ksa dks ns[kks vkSj izy;adkjh n`’;ksa dks Hkh ns[kksA ukuk

izdkj ds inkFkksZa dks ns[kksA muesa O;kIr lefefr vkSj lqUnjrk dks ns[kksA muls fc[kj jgs

jaxksa dks ns[kksA vius vklikl ds okrkoj.k dks /;ku ls ns[kks vkSj /;ku ls lquksA vkidks

vius okrkoj.k esa inkFkZ] ÅtkZ] dky vkSj vkdk’k i`Fkd&i`Fkd fn[kkbZ nsaxsA /;ku ls

ns[kus vkSj lquus ls vkidks vyx&vyx fn[kkbZ ns jgh ?kVukvksa esa varZlEcU/k feyrs

utj vkus yxsaxsA

18

/;ku ls ns[kuk] /;ku ls lquuk vkSj varlZEcU/kksa dks [kkstuk gh HkkSfrdh gSA ;g

lc ,d ftKklq gh dj ldrk gSA vr% ftKklq gksuk HkkSfrdh ds v/;;u gsrq tqVus vkSj

mlesa lQyrk ikus dh igyh 'krZ gSA

fdlh Hkh ?kVuk ds flyflysokj v/;;u ¼pkgs og lS)kafrd rjhds ls gks ;k

izk;ksfxd rjhds ls gks vFkok nksuksa gh rjhdksa ls gks½ ls le> iSnk gksrh gSA le> ls Kku

iSnk gksrk gSA rFkk Kku ls fo’ys"k.k djus] fu;a=.k djus] la’kksf/kr djus vkSj iwokZuqeku

yxkus dh {kerk fodflr gksrh gSA iwokZuqeku [kkst&dk;Z dks vkxs c<+krs gSa ftlls foKku

izxfr djrk gS vkSj lekt dks lEiUu cukus dk jkLrk feyrk gSA

HkkSfrdh esa dqN Hkh vafre fl)kar ds :i esa ugha gksrkA tks vkt lp dk vkHkkl

ns jgk gS] gks ldrk gS fd dy og xyr lkfcr gks tk;sA D;k vkidks dqN ;kn vk jgk

gS\ gk¡! ;kn dhft;s dksijuhdl] xSfyfy;ks] U;wWVu] Iykad] MkYVu] VkWelu] jnjQksMZ

vkfn dksA

HkkSfrdh esa dk;Z djus dh nks fn’kk;sa gSaA fdlh Hkh ?kVuk ;k fudk; dks Åij ls uhps ;k

uhps ls Åij tkrs gq, le>ukA bl rjg HkkSrdh esa fDy"V fudk; dks fgLlksaa esa ck¡Vuk

vkSj fQj fgLlksa dks le> dj iwjs fudk; dks le>uk gksrk gSA

HkkSfrdh esa v/;;u djrs gq, oSKkfudksa dks pkj ewyHkwr cyksa dh tkudkjh feyhA

nks nzO;&fi.Mksa ds chp fdz;k’khy cy dks xq:Rokd"kZ.k cy dgrs gSa pkgs os nzO;&fi.M

vfrlw{e d.k gksa vFkok fo’kkydk; czãk.Mh; fi.M gksaA nzO; esa ftu HkkSfrd vkSj

jlk;fud xq.kksa dh l`f"V gksrh gS] mlds ihNs ftl cy dk gkFk gksrk gS mls

fo|qrpqEcdh; cy dgrs gSaA fofHkUu rRoksa ds fuekZ.k ds ihNs fdz;k’khy cy dks ukfHkdh;

cy dgrs gSa tks nks izdkj ds gksrs gSaA vkt oSKkfud ml ewyHkwr ,dhd`r ije cy dks

tkuuk pkgrs gSa tks lHkh izdkj ds cyksa dh mRifRr dk dkjd gksA oSKkfudksa dh ;g

[kkst&;k=k ekuo tkfr ds dY;k.k ds fy;s u;s lk/ku tqVkus vkSj viuh Lo;a dh {kq/kk

dks 'kkar djus dk vuojr iz;kl gSA

HkkSfrdh ds v/;;u ls dbZ ,sls fu"d"kZ fudys gSa ftUgksaus rduhdh fodkl ds

jkLrs [kksys gSaA

rduhdh fodkl us u flQZ lekt ds fodkl esa viuh vge~ Hkwfedk fuHkkbZ gS oju~

HkkSfrdh o vU; {ks=ksa esa v/;;u gsrq u;s lk/ku Hkh tqVk;s gSaA vkt ,slk dksbZ Hkh {ks= ugha

gS tgk¡ HkkSfrdh dh enn u yh tkrh gksA vr% HkkSfrdh dk v/;;u dj fdlh Hkh {ks= esa

lQyrk dh uhao dks iq[rk fd;k tk ldrk gSA

19

2222---- izkphu Hkkjr esa ekiu& izkphu Hkkjr esa ekiu& izkphu Hkkjr esa ekiu& izkphu Hkkjr esa ekiu& fo'o ds izkphure xzFkksa esa osnksa dk uke mYys[kuh; gSA ;g gekjs izkphu Hkkjr esa

xkSjoiw.kZ dky dk nLrkost gSA ;g ns'k dh oSKkfud rFkk lkaLd`frd Å¡apkbZ;ksa dh >yd

nsrk gSA bldk izHkko rkRdkfyd laLd`fr;ksa] tSls fd felz] if'kZ;k rFkk vjc laLd`fr esa

fn[krk gSA

izkphu Hkkjr esa yEckbZ] le;] rkieku rFkk nzO;eku ds iw.kZfodflr ekiu ds

ek=d FksaA bldk laf{kIr ifjp; fuEukuqlkj gS&

2222----1 yEckbZ dk ek=d % ^n.M*1 yEckbZ dk ek=d % ^n.M*1 yEckbZ dk ek=d % ^n.M*1 yEckbZ dk ek=d % ^n.M*

_Xosn ¼4500&2500BC½ rFkk HkkLdjkpk;Z }kjk jfpr izkphu xzaFk ^^lw;Z

fl}kUr** ds vuqlkj izkphu Hkkjr esa iz;qDr gksus okys yEckbZ ds ek=d rFkk

O;qRiUu ek=d fuEu Fks

8 ijek.kq = 1 =ljs.kq

8 =ljs.kq = 1 js.kq

8 js.kq = 1 ckykxz

8 ckykxz = 1 fy[;

8 fy[; = 1 ;qd

8 ;qd = 1 ;o

8 ;o = 1 vaxqy

24 vaxqy = 1 gLr

4 gLr = 1 naM

2000 naM = 1 Øks'k

4 Øks'k = 1 ;kstu

_Xosn dh igyh _pk bl yEckbZ ds ek=d ^n.M* ¼tks fd yxHkx 1

ehVj yEckbZ esa Fkk½ dk i`Foh dh ifjf/k ls lEcU/k crkrh gS tks fd fuEu gS&

S = 1000X 2000 X 2000 X 24 vxaqy 25

= 24 X 109 X 1 n.M* 25 96

= 4 X 107 n.M

tks fd yxHkx i`Foh dh ifjf/k 4 X 107 ehVj ds cjkcj gSA vFkkZr n.M gekjs

vk/kqfud yEckbZ ds ek=d ehVj ls yxHkx leku FkkA

20

HkkLdjkpk;Z ¼ tUe 1114 bZ-½ }kjk jfpr izkphu xzaFk ^lw;Z fl}kr esa i`Foh dh

f=T;k vkB lkS ;kstu crk;h xbZ gS bl izdkj i`Foh dh ifjf/k

S = 2π X 800 ;kstu = 2π X 800 X 8000 n.M

= 4-02 X 107 n.M

gh izkIr gksrh gSA

rks bl lekurk ls ge ,d vaxqy dks 1@96 ehVj eku ldrs gS

1 vaxqy = 1-0416 ls-eh-

rFkk

,d ijek.kq = ¼8½&7 vaxqy

= 5 X 10&7 ls-eh-

= 5 A0

tks fd fdlh ijek.kq ds eki ds yxHkx cjkcj gSA

2222----2 nzO;eku dk ek=d % ^ek"k*2 nzO;eku dk ek=d % ^ek"k*2 nzO;eku dk ek=d % ^ek"k*2 nzO;eku dk ek=d % ^ek"k*

^pjd lafgrk* esa of.kZr nzO;eku ds ek=d fuEu FksA

10 d`".ky = 1 ek"k

10 ek"k = 1 lqo.kaZ

32 ek"k = 1 jkSI;@rkeziy = 10 ?kkj.k

48 ek"k = 1 ykSgiy = 10 eqf"V

300B.C. esa dkSfVY; us vius xzaFk ^vFkZ'kkL=*dkSfVY; us vius xzaFk ^vFkZ'kkL=*dkSfVY; us vius xzaFk ^vFkZ'kkL=*dkSfVY; us vius xzaFk ^vFkZ'kkL=* esa Hkkj ds ek=d dh lfoLrkj

foospuk dh gSA

2222----3 le; dk ek=d % ^gksjk*3 le; dk ek=d % ^gksjk*3 le; dk ek=d % ^gksjk*3 le; dk ek=d % ^gksjk*

izkphu Hkkjr esa ^lkou* ekiu i}fr le; dh x.kuk ds fy, izpfyr FkhA blds

ek=d o O;qRiUu ek=d Hkh iz;ksx esa FksA

60 foiy = 1 iy

60 iy = 1 ?kVh

2 ψ ?kVh = 1 gksjk 24 gksjk = 1 lkou fnu@lkSj fnol@u{k= fnu

^lw;Z fl}kUr* esa of.kZr ,d o"kZ dh vof/k 365¼ fnu dh ekuh xbZ gS

¼365 fnu@15 ?kVh@30 iy½ tks fd gekjs vk/kqfud o"kZ dh x.kuk ls esy [kkrh gSA

izkphu Hkkjr esa ^gksjk* dk vk/qkfud :i ^Hour* gSA

21

blds vfrfjDr lw;Z fl}kUr esa ukS vU; ekiu ds ek=d izpyu esa FksA ftuesa ls

fuEu izeq[k gS&

1½

1 czã o"kZ = 200 dYi

1 dYi = 1000 egk;qx

1 egk;qx = ¼4800$3600$2400$200½ fnO; o"kZ

= lR;qx $ =srk;qx$nzkij$dkfy;qx

= 12]000 fnO; o"kZ

= 4]32]000 lkSj o"kZ

2½ 1 fnO; o"kZ = 360 lkSj o"kZ

3½ 1 lkSj o"kZ = 365¼ lkou fnu ;k u{k= fnu

4½ 1 xkSjo = 60 lkSj o"kZ

5½ 1 izktkiR; ¼eUoUrj ½ = 71 egk;qx

6½ 1 fir`fnu = 1 pkUnzekl

2222----4 rkieku dk ek=d % ^fyad*4 rkieku dk ek=d % ^fyad*4 rkieku dk ek=d % ^fyad*4 rkieku dk ek=d % ^fyad*

oS'ksf"kd n'kZu esa ^Li'kZ* B.Mk rFkk xeZ dk ,glkl djkrk FkkA 'kCn ^Li'kZ*

vk/kqfud HkkSfrdh ds Temperature ¼rkieku½ ds leku FkkA m".krk dks ekius ds fy, ^

m".k Li'kZ* ek=d Fkk blesa ikuh ds teko fcUnq dks B.Mk ekurs gq, vf/kdre rki ikjs

ds ;k lksus ds xyu fcUnq dks ekurs FksA bu nksuksa vyx vf/kdre rki fcUnqvksa ij fuHkZj

nks ek=d izpyu esa Fks i) fyad ii) d{; 1 ;g nksuksa vkil esa fuEu izdkj lEcfU/kr

Fks&

1 izfyad = 1 iknd{; = 0.8850C

4 iknd{; = 1 d{; = 3.540C

6 d{; = 1 fyad = 21.240C

113 iknd{; = ikuh dk Hkki fcUnq ¼B.P.½ = 1000C

101 d{; = ikjs dk B.P. = 3570C

50 fyad = lksus dk xyu fcUnq ¼M.P.½ = 10620C

lUnHkZ iqLrds&lUnHkZ iqLrds&lUnHkZ iqLrds&lUnHkZ iqLrds&

1- Indian Heritage of Science and Technology The Physis by Dr. N.G. Dongre

2- Samskrita Bharti, New Delhi.

22

3- iz'kLrin Hkk"; rFkk U;k;dUnkyhiz'kLrin Hkk"; rFkk U;k;dUnkyhiz'kLrin Hkk"; rFkk U;k;dUnkyhiz'kLrin Hkk"; rFkk U;k;dUnkyh

4- Prashasta Pada Bhashya With Nyayakandali of Shridhara Pulised by E.J. Lazarus

and Co., Benares:- 1895

5- Kautileya Artha Shastra- (Ed) R.P. Kangle, Motilal Banarsidas Publishers Pvt.

Ltd. 1972.

6- Amshu Bodhine Of Maharishi Bhardwaja.

7- Surya Siddhanta (Ed) with Tattvamitra Sanskrit Commentary, Pub Jaiykrishna

das Haridas Gupta Benaras City-1946.

8- Vaisheshika Sidhantanam Ganitiya Paddhatya Vimmarshah-Dr. Narayan Gopal

Dongre, Pub., Director, Reseanch Instilute, Sampunanda Sanskrit Vishvavidyalya,

Varanas-1979.

9- Physics Ancient India (A Mathematical Analysis of Vaisheshida Philosophy), N.G.

Dongre Pub. Wiley Eastesn Limited (New Age Int. Pub. Ltd). 1995.

3333---- osx laLdkj& osx laLdkj& osx laLdkj& osx laLdkj& _f"k d.kkn }kjk ^oS'ksf"kd lw=* dh jpuk dh xbZ Fkh rFkk mlesa iz'kLrin Hkk";

esa *osx laLdkj* ¼Mechanical Force½ dk o.kZu gSA oS'ksf"kd fl}kUr esa of.kZr 'kCn ^deZ* dks lkekU; xkfr ¼Motion ½ ds fy, iz;ksx

fd;k x;k Fkk rFkk ^osx* 'kCn Force ¼cy½ ds vFkZ esa iz;qDr gksrk Fkk vFkkZr fdlh oLrq ij ^osx* ds iz;ksx ls ^deZ* izkIr gksrk FkkA ^xeudeZ* fdlh oLrq ds Momentum vFkkZr laosx dks n'kkZrh FkhA

Velocity ds fy, ^xfr* 'kCn dk iz;ksx gksrk Fkk ^js[kh; rFkk ^o`Ùkh; xfr* dk lfoLrkj mYys[k izkphu [kxksyfoKku ds xzUFkksa esa feyrk gSA ^fnd* 'kCn fn'kk dk Kku djkrh FkhA fnd~ lfn'k jkf'k Fkh rFkk Roj.k ¼Acceleration½ ds fy, ^osxp;* 'kCn iz;qDr fd;k tkrk FkkA

^osx laLdkj* dk o.kZu djus okys okD; ls fuEu xfr ds fu;e izkIr gksrs Fks 1- ^osx* ds iz;ksx ds ^deZ* mRiUu gksrk gSA

2- ^deZ* ds ifjorZu ^osx* ds lekuqikrh gksrs gS rFkk og osx dh fn'kk esa gksrs gSA

3- izR;sd la;ksx ¼Action½ ds fy, ,d fojks/kh rFkk cjkcj Reaction osx gksrk gSA vFkkZr nks oLrqvksa dk ,d nwljs ij ^osx* la;ksx cjkcj rFkk fojks/kh fn'kkvksa esa yxrk gSA

blh izdkj ge bl lUnHkZ xzaFk ls js[kh; xfr ds lHkh lw= izkIr dj ldrs gSA tSls fd v = u + at rFkk

v2= u2 + 2as

deZ Hkh pkj izdkj ds ekus x;s Fks

23

1111---- mR{ksi.k ¼mR{ksi.k ¼mR{ksi.k ¼mR{ksi.k ¼upward motion½½½½

2222---- vo{ksi.k ¼vo{ksi.k ¼vo{ksi.k ¼vo{ksi.k ¼downward motion½ ½ ½ ½

3333---- vkdqvkdqvkdqvkdqapu apu apu apu ¼¼¼¼motion due to tensile stress½½½½

4444---- izlkj.k ¼izlkj.k ¼izlkj.k ¼izlkj.k ¼Shear freed motion½ ½ ½ ½

4444---- fLFkfr LFkkidrk laLdkj ¼ fLFkfr LFkkidrk laLdkj ¼ fLFkfr LFkkidrk laLdkj ¼ fLFkfr LFkkidrk laLdkj ¼Elastic Forces½½½½ oS'ksf"kd n'kZu esa elasticity dks fLFkfrLFkkidrk dgk x;k gSA bldk o.kZu U;k;

dUnyh ds ,d 'yksd esa dqN bl izdkj gS& rkieku dh rjg Bksl oLrqvksa esa fLFkfrLFkkidrk dk Hkh xq.k gksrk gS rFkk bl xq.k

ds dkj.k Bksl oLrqvksa esa] cy yxkus ij tks fo:i.k gksrk gS mlls] okfil viuk vkdkj izkIr dj ysrh gSA

djdkoyh esa elasticity ¼izR;kLFkrk½ dks fuEu izdkj le>k;k x;k gS& elastic forces Bkslksa rFkk vn`'; inkFkksZ esa mRiUu gksrs gS rFkk blh dkj.k inkFkZ

Vibrate ¼dfEir½ gksrs gSA 5555---- xq:Ro rFkk xq:Rokd"kZ.k&xq:Ro rFkk xq:Rokd"kZ.k&xq:Ro rFkk xq:Rokd"kZ.k&xq:Ro rFkk xq:Rokd"kZ.k& oS'ksf"kd n'kZu ds vuqlkj xq:Ro gh Bksl rFkk nzO; dk uhps dh vksj fxjus dk

dkjd gSA ;g vn`'; gS rFkk inkFkksZ ds fxjus ls gh bldk Hkku gksrk gSA rFkk xq:Ro dk Kku Bkslksa dh Hkhrjh vkUrfjd d.kksa ds eki ij fuHkZj gSA

^U;k;dUnyh* ds o.kZu ds vuqlkj xq:Ro ds dkj.k Bkslksa dk izR;sd d.k ,dleku :i ls uhps fxjrk gSA vFkkZr ,d cM+h oLrq rFkk ,d NksVh oLrq ,dleku :i ls xq:Ro ds dkj.k uhps fxjrh gSA

tcfd if'peh ns'kksa esa vjLrq ds fl}kUr ls oLrqvksa ds uhps fxjus dh nj dk oLrqvksa ds Hkkj ds lekuqikrh ekuk tkrk Fkk vFkkZr Hkkjh oLrq tYnh uhps fxjsxh rFkk gYdh oLrq nsj lsA lu~ 1590 esa xSfyfy;ksa us ;g fl) fd;k fd lHkh oLrq, xq:Ro ds dkj.k ,d leku xfr ls uhps fxjrh gS] exj ;g fl}kUr izkphu Hkkjr esa _f"k d.kkn_f"k d.kkn_f"k d.kkn_f"k d.kkn ds oS'ksf"kd n'kZu esa ekStwn FkkA

,d vU; 'yksd esa ;g crk;k x;k gS fd i`Foh dh lrg ij izR;sd oLrq i`Foh ds dsUnz dh vksj tkrh gqbZ js[kk dh fn'kk esa fxjrh gSA

HkkLdjkpk;Z ¼1134 bZ-½ ds vuqlkj i`Foh esa viuh vksj [khapus dh {kerk gS rFkk ,d Hkkjh fi.M ¼xq:½ tks czãekaM esa mifLFkr gS vius Hkkj ds lekuqikr ¼xq:Ro½ esa i`Foh dks [khprk gSA

vFkkZr czgekaM dh izR;sd oLrq ,d nwljs dks viuh vksj [khaprh gSA blh izdkj [kxksyh; fi.Mksa dh xfr foKku rFkk mudh lVhd x.kuk,a vusd izkphu xzaFkksa esa mYysf[kr gSA lanHkZ iqLrd&Physic in Ancient India (A Mathematical Analysis of Vaisheshika

Philosophy)- N.G. Dongre Pub., Wiley Eastern Limited (New Age Publishers Limited)-1995.

24

AA →

lfn’k dks n’kkZuk

A→

Z

X

YO

Ax

Az

Ay

lfn’k dk fo;kstu

6666---- vfn’k vkSj lfn’kvfn’k vkSj lfn’kvfn’k vkSj lfn’kvfn’k vkSj lfn’k (Scalars and vectors)&&&&

¼1½ ¼1½ ¼1½ ¼1½ vfn’k vkSj lfn’kvfn’k vkSj lfn’kvfn’k vkSj lfn’kvfn’k vkSj lfn’k& & & & dqN HkkSfrd jkf’k;kWa tSls le; (time)] nzO;eku (mass)] Å"ek (heat)] rkiØe (temperature) vkfn ,slh jkf’k;kWa gSa ftudh vfHkO;fDr ds fy;s fn’kk ds Kku dh vko’;drk ugha gksrh ,oa fdlh ,d vad ek= ls gh budks iwjh rjg ls vfHkO;Dr fd;k tk ldrk gSA ,slh jkf’k;kWa vfn’k (scalars) dgykrh gSaA ysfdu dqN HkkSfrd jkf’k;kWa tSls foLFkkiu (displacement)] osx (velocity)] laosx (momentum)] cy (force) vkfn ,slh Hkh gksrh gSa ftudks fcuk fn’kk ds Kku ds iwjh rjg ls vfHkO;Dr ugha fd;k tk ldrk gSA ,slh jkf’k;ksa dks lfn’k (vectors) dgrs gSaA

lfn’k jkf’k dks ge ;k rks cksYM o.kZ (bold latters) ds :Ik esa ;k fQj o.kZ ds Åij ,d rhj (arrow) [khap dj fy[krs gSaA o.kZ lfn’k ds ifjek.k dks n’kkZus ds fy;s gksrk gS tcfd cksYMusl vFkok ml ij ,sjks dk fpUg mldh fn’kk ls lacaf/kr jgrk gSA lfn’k jkf’k dks xzkQh; rjhds ls vfHkO;Dr djus ds fy;s ge ,d funsZ’k ra= dk lgkjk ysrs gSaA blesa lfn’k jkf’k ds ijhek.k (magnitude) ds cjkcj ,d js[kk [khaprs gSa rFkk ftl fn’kk esa HkkSfrd jkf’k fØ;k’khy gksrh gS ml fn’kk dh vksj js[kk ds fljs ij ,d rhj dk fpUg cuk nsrs gSaA bls ^osDVj ;k lfn’k^ ls lacksf/kr djrs gSaA fdlh Hkh lfn’k ( )A

r dks ml funsZ’k

ra= (frame of reference) dh v{kksa ¼tSls x-, y-,z-½ ds lkis{k fo;ksftr dj ds mlds lfn’k ?kVd ( )zyx AA,A

rrr izkIr dj ldrs

gSaA vr% lfn’k ?kVdksa ds :Ik esa lfn’k ( )Ar

dks ge fuEukuqlkj fy[krs gSa%

zyx AAAArrrr

++= ..(A)

ifjHkk"kk ds vuqlkj izR;sd lfn’k ?kVd funsZ’k ra= (frame of reference) dh ,d v{k dh vksj jgrk gSA vxj ge funsZ’k ra= (frame of reference) dh v{kksa ¼tSls x-, y-, z-½ ds lkis{k bdkbZ lfn’k ¼bdkbZ lfn’k os gksrs gSa ftudk ijhek.k bdkbZ ek=d gksrk gS vkSj os fdlh fufnZ"V fn’kk esa gksrs gSa½ j,i ˆˆ

vkSj k dks ifjHkkf"kr djsa rks ge xAr

dks iAxˆ vkSj blh rjg yA

r rFkk zA

r dks Hkh fy[k ldrs gSaA

vr% lfn’k ( )Ar

kAjAiAA zyxˆˆˆ ++=

r ... (B)

;gkWa zyx A,A,A vad gSaA bUgsa Ar lfn’k ds ?kVd dgrs gSaA vxj ge x y- ry esa fLFkr fdlh

lfn’k ij fopkj djsa vkSj ekusa fd lfn’k ( )Ar

/kukRed x- v{k ls θ dks.k cukrk gSs ¼;kn jgs ;gkWa /kukRed x- v{k ls dks.k θ dks ekik tk jgk gSA _.kkRed x- v{k ls θ dks.k dk eryc /kukRed x- v{k ls ¼π−θ½ dks.k gksxk½ rks gesa blds fuEu nks ?kVd ( )yx A,A feysaxs%

AsinθA AcosθA yx == ... (C)

25

A

A

ΒΒΒΒθ

ΒΒΒΒ

fp=&3% xq.ku izfØ;k

ΒΒΒΒ

A cosθ B

AΒΒΒΒcosθA

fp=&4% B dk izkstsD’ku A ij rFkk

A dk izkstsD’ku B ij

ik;Fkkxksjl cks/kk;u izes; ls lfn’k dk ijhek.k%

( ) ( )2y

2x AAA += ... (D)

rFkk] f=dks.kferh (Trigonometry) dh lgk;rk ls lfn’k dh x- v{k ds lkis{k fn’kk fuEu O;atd ls izkIr dh tk ldrh gS %

x

y

A

Atanθ = ;kfu]

x

y

A

Aθ

1tan−= ... (E)

(2) MkWV xq.kuMkWV xq.kuMkWV xq.kuMkWV xq.ku (Dot product) :

fdlh fudk; ij dksbZ cy vkjksfir djus ls Tkc og dqN nwjh ij foLFkkfir gks tkrk gS rks ge dgrs gSa fd geus dk;Z fd;kA dk;Z dh x.kuk ds fy;s gesa ekywe gksuk pkfg;s fd geus cy fdruk vkSj fdl fn’kk esa yxk;k rFkk gesa foLFkkiu fdruk vkSj fdl fn’kk esa feykA cy rFkk foLFkkiu nksuksa lfn’k gSa vkSj buesa xq.kk djus ls gesa dk;Z dk ijhek.k izkIr gksrk gSA ysfdu] dk;Z dks n’kkZus ds fy;s gesa fn’kk ugha crkuk iM+rhA dk;Z ,d vfn’k jkf’k gSA bl rjg ge ns[krs gSa fd ;gkWa nks lfn’kksa A

r rFkk B

resa xq.kk djus ls gesa vfn’k ds :Ik esa

ifj.kke (resultant) feyrk gSA bl xq.ku&izfØ;k dks vfHkO;Dr djus ds fy;s ge Ar rFkk B

r ds

chp MkWV ( )⋅ dk iz;ksx djrs gSaA vr% MkWV xq.ku nks lfn’kksa ds chp gksus okyk ,d ,slk xq.kk gS ftldk ifj.kke gesa ges’kk ,d vfn’k ds :Ik esa feyrk gSA ;gh dkj.k gS fd bls ge vfn’k xq.ku Hkh dgrs gSaA

vfn’k xq.ku dks izkIr djus ds fy;s ge nksuksa lfn’k dks bl rjg [khaprs gSa fd muds fljs ,d fcanq ij vk tk;saA ekuk fd bl le; buds chp esa θ dks.k gSA ¼;g dks.k lnSo 0 ls 180o ds chp esa jgrk gSA½ vc ge vxj fp= dks ns[ksa rks ik;saxs fd lfn’k B

r dk lfn’k A

r ij izkstsD’ku

θcosB gS vkSj ;g Ards lekarj gSA vfn’k xq.kk dks ge

lfn’k Ar

ds ijhek.k vkSj lfn’k Br

ds Ar

ij izkstsD’ku θcosB ds xq.kuQy ds :Ik esa ifjHkkf"kr djrs gSaA

)A(BcosBA θ=⋅rr

… (A)

vfn’k xq.ku dks ge lfn’k Br ds ijhek.k vkSj lfn’k A

r ds B

r ij izkstsD’ku

θcosA ds xq.kuQy ds :Ik esa Hkh ifjHkkf"kr dj ldrs gSaaA vr%

( )θAcosBAB =⋅rr

mi;qZDr laca/kksa ls ge ns[krs gSa fd vfn’k xq.kk esa Ar rFkk B

r vkil esa Øefofues; (commute)

djrs gSaA ;kfu]

ABBArrrr

⋅=⋅ … (B)

Ar rFkk B

r dks ?kVd ds :Ik esa fy[kus ij

zzyyxx BABABABA ++=⋅rr

… (C)

vxj ge j,i ˆˆ vkSj k tks fd x, y rFkk z fn’kk esa bdkbZ lfn’k gSa] ds fy;s fopkj djsa rks gesa fuEu mi;ksxh O;atd feyrs gSa%

26

Thumb

FingersBA

Thumb

Fingers

BA

lfn’k xq.ku izfØ;k esa fn’kk dk laKku

A

ΒΒΒΒθ

n

A ΒΒΒΒ

fp=&5% lfn’k xq.ku izfØ;k

1kkjjii =⋅=⋅=⋅ ˆˆˆˆˆˆ

0ikkjji =⋅=⋅=⋅ ˆˆˆˆˆˆ

(3) ØkWl xq.kuØkWl xq.kuØkWl xq.kuØkWl xq.ku (Cross product) :

tc fdlh fudk; ij dksbZ leku ijhek.k okys cy ,d lkFk ,d gh fcanq ij foijhr fn’kkvksa esa fØ;k’khy gksrs gSa rks og fudk; lkE;koLFkk esa iM+k jgrk gSA ysfdu tc ;s gh cy vyx&vyx fcanqvksa ij vkjksfir gksrs gSa rks fudk; fdlh v{k ds ifjr% ?kweus dk iz;kl djrk gSA ;g ?kqeuk fudk; ij vkjksfir cy vkSj muds chp fo|eku yEcor nwjh ij fuHkZj djrk gSA fudk; dk ?kqeuk nks izdkj ls gks ldrk gSA og ;k rks ?kM+h dh fn’kk esa gksrk gS vFkok mldh foijhr fn’kk esaA ?kqeus ds fy;s vko’;d HkkSfrd jkf’k cy vk?kw.kZ (torque) dgykrh gSA ;g ,d lfn’k jkf’k gSA vk?kw.kZ dh x.kuk ds fy;s gesa cy rFkk muds chp dh yEcor nwjh dk vkil esa xq.kk djrs gSaA bl rjg ge ns[krs gSa fd ;gkWa nks lfn’kksa esa xq.kk djus ij gesa lfn’k ds :Ik esa ifj.kke feyrk gSA bl xq.ku izfØ;k dks vfHkO;Dr djus ds fy;s ge A

r rFkk B

r ds chp

ØkWl ( )× dk iz;ksx djrs gSaA vr% ØkWl xq.ku nks lfn’kksa ds chp gksus okyk ,d ,slk xq.kk gS ftldk ijh.kke gesa ges’kk ,d lfn’k ds :Ik esa feyrk gSA ;gh dkj.k gS fd bls ge lfn’k xq.ku Hkh dgrs gSaA

Ar

rFkk Br

ds chpa lfn’k xq.ku dks izkIr djus ds fy;s ge nksuksa lfn’kksa dks bl rjg [khaprs gSa fd muds fljs ,d fcanq ij vk tk;saA ;s lfn’k ,d ry esa fLFkr gksaxsA ekuk fd bl le; buds chp esa θ dks.k gSA lfn’k xq.ku BA

rr× dks ge ,d lfn’k ds :Ik esa ifjHkkf"kr djrs gSa ftldk ijhek.k θsinAB

gksrk gSA T;kWferh;&:Ik esa ;g eku lekarj prqHkqZt (parallelogram) ds ml {ks=Qy ds cjkcj gksrk gS ftldh Hkqtk,Wa A

r rFkk B

r gksrh gSaA BA

rr× dh fn’kk ml ry ds yEcor gksrh gS ftlesa A

r vkSj

Br mifLFkr jgrs gSaA bl ry ds yEcor nks fn’kk;sa gks ldrh gSaA vr% BA

rr× dh fn’kk dks izkIr

djus ds fy;s ge θ dks Ar ls B

r dh vksj ¼nks laHkkfor NksVs dks.kksa esa ls vis{kkd`r NksVs dks.k dks

cukrs gq,½ /kukRed ysrs gq, ekirs gSaA BArr

× dh fn’kk izkIr djus ds fy;s ge nkfgus gkFk ds LØw (right handed screw) dk iz;ksx dj ldrs gSaA A

r ls B

r dh vksj ?kqekus ij nkfgus gkFk dk LØw

ftl fn’kk esa vkxs c<+rk gS] og BArr

× dh fn’kk dgykrh gSA ;kn jgs fdtc ge Brls A

r dh vksj

LØw dks ?kqekrs gSa rks ;g ihNs vkrk gSA vr% BArr

× vkSj ABrr

× dh fn’kk,Wa ,d nwljs ds foijhr gksrh gSaA

ABBArrrr

×−=× ... (A)

bl rjg ABBArrrr

×≠× gS ftlls gesa irk pyrk gS fd lfn’k xq.kk esa A

r rFkk B

r vkil esa Øe fofue; (commute) ugha

djrs gSaA

BArr

× dh fn’kk izkIr djus ds fy;s ge vius nkfgus gkFk dh maxfy;ksa dks eqDds dh 'kDy esa bl rjg j[ksa fd gekjk vaxwBk ckgj dh vksj fudyk jgsA maxfy;ksa ds fljs A

r ls B

r dh

vksj ?kqekus dh fn’kk vkSj vaxwBk BArr

× dh fn’kk crk;sxkA

ABsinθBA =×rr

n .. .(B)

;gkWa n ,d bdkbZ lfn’k gS ftldh fn’kk ml ry ds yEcor gksrh gS ftlesa A

r vkSj B

r fLFkr gksrs gSaA

27

nksuksa lfn’kksa ds chp ds dks.k θ ds 0 vFkok 180o gksus ij BArr

× dk eku 'kwU; gksrk gSA ;kfu] nks lekarj vFkok foijhr lekarj lfn’kksa dk xq.kuQy ges’kk 'kwU; gksrk gSA ysfdu] tc ;s nksuks lfn’k vkil esa yEcor gksrs gSa rc gesa BA

rr× dk vf/kdre eku feyrk gSA

?kVd ds :Ik esa fy[kus ij

zyx

zyx

BBB

AAA

kji

BA

ˆˆˆ

=×rr

… (C)

vxj ge Ar rFkk B

r dks x, y rFkk z fn’kk esa bdkbZ lfn’k ekusa rks gesa fuEu mi;ksxh O;atd feyrs

gSa%

0kkjjii =×=×=× ˆˆˆˆˆˆ … (D)

rFkk]

jik,ikj,kji ˆˆˆˆˆˆˆˆˆ =×=×=× … (E)

Elementary formulation of differential and integral Calculus

Actually these are the calculus of infinitesimal. These are the tools to handle the physical quantities that vary with respect to some variable.

Here the lines, surfaces and volumes are divided into a large number of very small parts. Then one considers the interrelations in the limiting case when size of each subdivision goes to zero

Differential Calculus

Let us consider a physical quantity, say displacement, changes in regular way as the value of time ‘t’ changes. Case I: displacement is proportional to time x(t) = At

Let us consider displacements at two moments of time one at ‘t’ and other at ‘t+∆t’ x(t) = A t x(t+∆t) = A (t+∆t)

the difference in displacements i.e. the distance travelled in time ‘∆t’ is

A (t+∆t) – A t = A (∆t)

On dividing by ∆t, we get exactly A. This is what is known as differential coefficient of At with respect to ‘t’.

Case II: displacement is proportional to square of time x(t) = At2 Let us consider displacements at two moments of time one at ‘t’ and other at ‘t+∆t’ x(t) = A t2 x(t+∆t) = A (t+∆t)2

The difference in displacements i.e. the distance travelled in time ‘∆t’ is

A (t+∆t)2 – A t2 = 2A t (∆t) + (∆t)2

On dividing by ∆t, we get 2At + ∆t. when ∆t is made infinitesimal, it can be neglected and we get differential coefficient of At2 with respect to ‘t’ as 2At.

28

In general differential coefficient of any function f(x) with respect to ‘x’ or the rate of change of f(x) with respect to ‘x’ is defined as

h

xfhxfh

dx

xdf )()(0lim

)( −+→=

Double and higher derivatives

Double derivative: The rate of change of dx

xdf )(with respect to ‘x’ is

=dx

xdf

dx

d

dx

xfd )()(2

2

Triple derivative: The rate of change of 2

2 )(dx

xfdwith respect to ‘x’ is

= 2

2

3

3 )()(dx

xfd

dx

d

dx

xfd

Similarly the differential coefficient of any function x(t) with respect to ‘t’ or the rate of change of x(t) with respect to ‘t’ is defined as

t

txttxt

dt

dx

∆−∆+→∆= )()(

0lim

If the function is ‘displacement’, we call this as velocity. Further, the rate of change of velocity is acceleration. The use of calculus is of great advantage in evaluating the motion related parameters.

Second equation of motion

s = ut +1/2 at2

On differentiating ‘s’ with respect to time, we get velocity ‘v’.

atudt

dsv +==

On differentiating ‘v’, we get acceleration ‘a’

2

2

dt

sd

dt

dva ==

Integral calculus:

We have seen above that the differential calculus considers relation between parts of geometrical figures, when the parts become infinitesimal.

The task of integral calculus s entirely opposite. It is to produce geometrical figures of final size by integration of infinitesimal parts.

∫∫ == xdxdtdt

dx

The symbol used in front of the integrand is nothing but an elongated ‘s’ standing for the word ‘sum’.

29

The integration of acceleration with respect to time means the result will be average velocity during the given time.

∫∫ === atadtdtdt

xd

dt

dx2

2

= v

The integration of velocity with respect to time means the result will be total displacement during the given time.

∫ ∫∫ ===2

2taatdtvdtdt

dt

dx= s

If v is constant,

∫ == svtvdt

The area under the velocity curve gives total displacement during the given time.

The area under the acceleration curve gives average change in velocity during the given time.

Definite integrals:

If the limits of integration are provided, then we write the limits over ingral symbol. The integration of (1) with respect to 'x' within limits x=0 to x=9 is written as and the value after integration is obtained as

9099

0

9

0

=−==∫=

=

xdxx

x

Let the integration of ( )xf with respect to 'x' is g(x). This integration within limits x=0 to x=9 is written as and the value after integration is obtained as

( ) ( ) ( ) ( )099

0

9

0

ggxgdxxfx

x

−==∫=

=

The integration of sinx within limits x = 0 to x = 2π is

( ) 00cos2coscossin2

0

2

0

=−−=−=∫ πππ

xxdx

Use of calculus

In order to use integral calculus, we need to divide the given system into infinitesimal suitable parts. After this we are to find the effect of our interest due to one such infinitesimal part. Then the effects due to all parts are integrated.

Line is divided into segments Plane is divided into strips Disc and shells are divided into rings Sphere is divided into shells

Generalization: The above elementary formulation based on calculum can be extended to two and three dimensions.

f}in izes;f}in izes;f}in izes;f}in izes;

30

HkkSfrdh esa lfUudVuksa dk cgqr egRo gSA lfUudVuksa dk iz;ksx djrs le; gesa O;atd esa iz;qDr izR;sd jkf'k;ksa ds ifj.kkeksa ij fopkj djuk gksrk gSA vxj gesa fdlh O;atd esa ,d jkf'k 1000 gS rFkk nwljh 1 gS rks ge 1000 dh rqyuk esa 1 dks ux.; eku dj NksM+ ldrs gSaA ysfdu 10 dh rqyuk esa 1 dks ux.; ugha ekuk tk ldrkA ekukfd gekjs ikl ,d O;atd gS% (1+x)n

ekuk fd 1 dh rqyuk esa x cgqr NksVk gS vc vxj ge (1+x)n dk foLrkj djsa rks

( ) ( ) ( )( ).......

!3

21

!2

111 32 x

nnnx

nnnxn n −−+−++=+

vc pw¡fd x cgqr NksVk gS rc x2, x3 …… x dh rqyuk esa vR;Ur gh NksVs gksaxs vkSj mUgsa ux.; ekurs gq, NksM+k tk ldrk gSA vr% HkkSfrdh; egRo dks /;ku esa j[krs gq, ( )nx+1 dks ( )nx+1 fy[kk tk ldrk gSA ,slk djus ij ?kVuk ds v/;;u ij xq.kkRed n`f"V ls dksbZ izHkko ugha iM+rk gSA

19

2222 ---- i zk p hu H k k jr e sa eki u & i zk p hu H k k jr e sa eki u & i zk p hu H k k jr e sa eki u & i zk p hu H k k jr e sa eki u &

fo'o d s i zkp hur e xzFk ksa e sa o sn ksa dk u ke m Y ys[ ku h; gSA ; g gekjs i z kp hu Hkkjr

e sa xkS joi w. kZ d ky d k n Lr ko st gSA ; g n s'k d h o SK kfud r Fkk l ka Ld `fr d Å ¡ap kb Z; ksa d h

> yd n sr k gSA b l d k i zH kko r kR d kfyd l a L d `fr ; ksa] t Sl s fd fel z ] i f'kZ; k r Fkk v jc

l a Ld `fr esa fn [kr k gSA

izkp hu H kkjr esa yEc kb Z] l e; ] r ki e ku r Fkk n zO ; e ku d s i w. kZfod flr ekiu d s

e k= d FksaA b l dk l af{ kI r i fjp; fu Eu ku qlkj gS&

2222 ---- 1 y Ec kb Z d k e k= d % ^n . M*1 y Ec kb Z d k e k= d % ^n . M*1 y Ec kb Z d k e k= d % ^n . M*1 y Ec kb Z d k e k= d % ^n . M*

_ X o sn ¼4 50 0 & 25 0 0BC½ r Fkk H kk Ld jkp k; Z } kjk j fp r i zkp hu xzaFk ^^ l w; Z

fl } kUr ** d s v u ql kj i zkp hu H kk jr esa i z; qDr gksu s o kys yEc kb Z d s e k= d r Fkk

O; qRi Uu ek=d fu Eu Fks

8 i j ek. kq = 1 = ljs. kq

8 = l js. kq = 1 js. kq

8 j s. kq = 1 c ky kxz

8 c ky kxz = 1 fy[ ;

8 fy[; = 1 ; qd

8 ; qd = 1 ; o

8 ; o = 1 v axqy

24 v axqy = 1 gLr

4 gLr = 1 n aM

20 0 0 n aM = 1 Ø ks' k

4 Ø ks'k = 1 ; kstu

_ X o sn d h i gy h _ p k b l yEc kb Z d s ek=d ^n. M* ¼t ks fd yxH kx 1

ehV j yEc kb Z esa Fkk½ d k i `Fo h d h i fjf/ k l s l Ec U/ k c r kr h gS t ks fd fu Eu gS&

20

S = 1 0 0 0X 20 0 0 X 2 0 00 X 2 4 v xa qy 2 5

= 2 4 X 109 X 1 n. M* 2 5 9 6

= 4 X 10 7 n . M

tks fd yx Hkx i `Fo h d h i fjf/ k 4 X 1 0 7 ehV j d s c jkc j gSA vFkkZr n. M gekjs

v k/kqfu d yEc kb Z d s e k= d e hV j ls yxHk x le ku FkkA

HkkL d jkp k; Z ¼ tUe 1 1 14 b Z- ½ } kj k jfp r i zkp hu xzaFk ^l w; Z fl} kr e sa i `Fo h d h

f=T; k v kB l kS ; kstu c r k; h xb Z gS b l i zd kj i `Fo h d h i fjf/ k

S = 2π X 800 ; kst u = 2π X 800 X 8000 n. M

= 4 - 02 X 107 n. M

gh i zkIr gks r h gSA

r ks b l l ekur k l s ge , d v axqy d ks 1 @9 6 ehV j e ku l d r s gS

1 v axqy = 1 -0 4 16 ls- eh-

r Fkk

,d i j ek. kq = ¼8 ½&7 v axqy

= 5 X 10&7 l s- eh-

= 5 A0

tks fd fd l h ij ek. kq d s eki d s yxH kx c jk c j gSA

2222 ---- 2 n zO ; e ku d k e k=d % ^e k" k*2 n zO ; e ku d k e k=d % ^e k" k*2 n zO ; e ku d k e k=d % ^e k" k*2 n zO ; e ku d k e k=d % ^e k" k*

^p jd l afgr k* esa of. kZr n zO ; eku d s ek= d fu Eu FksA

10 d `" . ky = 1 ek" k

10 e k" k = 1 l qo. ka Z

32 e k" k = 1 jkSI; @ r kezi y = 10 ?kkj . k

48 e k" k = 1 ykSgi y = 1 0 eqf" V

30 0B.C. esa d kS fV Y ; u s v i u s x zaFk ^v F kZ 'k kd kS fV Y ; u s v i u s x zaFk ^v F kZ 'k kd kS fV Y ; u s v i u s x zaFk ^v F kZ 'k kd kS fV Y ; u s v i u s x zaFk ^v F kZ 'k k L= *L= *L= *L= * esa H kkj d s e k= d d h l fo Lr kj

fo o spu k d h gSA

21

2222 ---- 3 l e ; d k e k= d % ^g ksj k*3 l e ; d k e k= d % ^g ksj k*3 l e ; d k e k= d % ^g ksj k*3 l e ; d k e k= d % ^g ksj k*

izkp hu Hkkjr esa ^lkou* e ki u i} fr le; d h x. ku k ds fy, i zp fy r FkhA b l ds

e k= d o O; qRi Uu e k= d H kh i z; ksx e sa FksA

60 foi y = 1 i y

60 i y = 1 ?kV h

2 ψ ? kV h = 1 gksjk 24 gksjk = 1 l kou fnu @ l kSj fno l @u{ k= fn u

^ l w; Z fl } kUr * e sa of. kZr , d o" kZ d h v of/ k 3 6 5¼ fn u d h e ku h xb Z gS

¼3 65 fnu @1 5 ?kV h@3 0 i y½ t ks fd ge kjs vk/ kqfu d o" kZ d h x. ku k ls e sy [ kkrh

gSA

i zkp hu H kkjr e sa ^gksjk* dk v k/qkfu d : i ^Hour* gSA

b l d s v fr fj Dr lw; Z fl } kUr e sa u kS vU; eki u d s ek=d i zp yu esa FksA

ft u e sa ls fu Eu i ze q[k gS&

1½

1 c zã o" kZ = 20 0 d Yi

1 d Y i = 10 0 0 egk; qx

1 egk; qx = ¼4 80 0$ 3 6 00 $ 2 40 0 $2 00½ fn O ; o" kZ

= lR; qx $ = sr k; qx$ n zki j$ d kfy; qx

= 12 ]0 0 0 fn O; o" kZ

= 4] 32 ]0 00 lkS j o" kZ

2½ 1 fn O; o" kZ = 36 0 l kS j o"kZ

3½ 1 l kS j o" kZ = 36 5¼ l kou fnu ; k u { k= fnu

4½ 1 xkS jo = 60 l kSj o" kZ

5½ 1 i zkt ki R; ¼eUoUr j ½ = 7 1 egk; qx

6½ 1 fir `fn u = 1 p kUn zekl

2222 ---- 4 r ki e ku d k e k= d % ^ fy ad *4 r ki e ku d k e k= d % ^ fy ad *4 r ki e ku d k e k= d % ^ fy ad *4 r ki e ku d k e k= d % ^ fy ad *

o S'ksf" kd n 'kZu esa ^ Li 'kZ* B . Mk rFkk xe Z d k , glkl d jkr k FkkA 'kC n

^ Li 'kZ* v k/ kqf u d H kkSfr d h d s Temperature ¼r ki e ku ½ d s l eku FkkA m" . kr k d ks eki u s ds

22

fy, ^ m ". k Li 'kZ* ek= d Fkk b le sa i ku h ds te ko fc Un q d ks B. Mk eku rs gq, v f/ kd r e

r ki i kjs d s ; k l ksu s d s xyu fc Un q d ks e ku r s FksA b u n ksu ksa v yx v f /kd r e r ki

fc Un qvksa i j fu H kZj n ks e k= d i zp yu e sa Fks i) fyad ii) d {; 1 ; g n ksu ksa v ki l e sa fu Eu

i zd kj lEc f U/ kr Fks&

1 i zfy ad = 1 i kn d{ ; = 0.8850C

4 i kn d{; = 1 d { ; = 3.540C

6 d { ; = 1 fyad = 21.240C

11 3 i kn d {; = i ku h d k H kk i fc Un q ¼B.P.½ = 1000C

10 1 d {; = i kjs d k B.P. = 3570C

50 fyad = l ksu s d k xyu fc Un q ¼M.P.½ = 10620C

l Un H kZ i qLr d s&l Un H kZ i qLr d s&l Un H kZ i qLr d s&l Un H kZ i qLr d s&

1 - Indian Heritage of Science and Technology The Physis by Dr. N.G. Dongre

2 - Samskrita Bharti, New Delhi.

3 - iz' kLr in Hkk" ; r Fkk U; k; d Un ky h

4 - Prashasta Pada Bhashya With Nyayakandali of Shridhara Pulised by E.J.

Lazarus and Co., Benares:- 1895

5 - Kautileya Artha Shastra- (Ed) R.P. Kangle, Motilal Banarsidas Publishers Pvt.

Ltd. 1972.

6 - Amshu Bodhine Of Maharishi Bhardwaja.

7 - Surya Siddhanta (Ed) with Tattvamitra Sanskrit Commentary, Pub Jaiykrishna

das Haridas Gupta Benaras City-1946.

8 - Vaisheshika Sidhantanam Ganitiya Paddhatya Vimmarshah-Dr. Narayan Gopal

Dongre, Pub., Director, Reseanch Instilute, Sampunanda Sanskrit

Vishvavidyalya, Varanas-1979.

9 - Physics Ancient India (A Mathematical Analysis of Vaisheshida Philosophy),

N.G. Dongre Pub. Wiley Eastesn Limited (New Age Int. Pub. Ltd). 1995.

3333 ---- o sx l a Ld k j& o sx l a Ld k j& o sx l a Ld k j& o sx l a Ld k j&

_ f" k d . kkn } kjk ^ oS'ksf" kd l w = * d h jp u k d h xb Z F kh r Fk k m le sa i z' kLr in H kk" ;

e sa * o sx l a Ld kj * ¼Mechanical Force½ d k o. kZu gSA

23

oS' ksf" kd fl } kUr esa of.kZr 'kC n ^de Z* d ks l kek U; xkfr ¼Motion ½ d s fy, i z; ksx

fd ; k x; k Fkk r Fk k ^ o sx* ' kC n Force ¼c y½ d s vFkZ esa i z; qDr gksr k Fkk v FkkZr fd l h o Lr q

i j ^ o sx* d s i z; ksx ls ^d eZ* i zkI r gksr k FkkA ^ xe u d eZ* fd l h o Lr q d s Momentum v Fkk Zr

l a o sx d ks n'kkZr h Fk hA

Velocity d s fy, ^xfr * 'kC n d k i z; ksx gksrk Fkk ^ js[ kh ; r Fkk ^ o ` Ùkh; x fr * d k

l fo Lr kj mY ys[ k i zkp hu [ kxksyfo K ku d s xzUFkksa e sa fey r k gSA ^fn d * 'kC n fn 'kk d k K ku

d jkr h FkhA fn d ~ l fn 'k jk f'k F kh r Fkk R o j. k ¼Acceleration½ d s fy, ^ o sxp ;* 'kC n

i z; qDr fd ; k t kr k FkkA

^ osx l a Ld kj* dk o. kZu d ju s o ky s o kD ; ls fuEu xfr d s fu ; e i zkI r gksr s Fks

1- ^ o sx* d s i z;ksx ds ^d e Z* mRi Uu gksr k gSA

2- ^d eZ* d s i fjo r Zu ^o sx* d s le ku qikr h gksr s gS r Fkk og o sx d h fn 'kk e sa gksr s

gSA

3- i zR; sd la; ksx ¼Action½ d s fy, , d fo j ks/ kh r Fkk c jkc j Reaction o sx gksr k

gSA v FkkZr n ks o Lr qvksa d k , d n w l js i j ^ o sx* la; ksx c jkcj r Fkk fo j ks/ kh

fn 'kkv ksa e sa yxr k gSA

b l h i zd kj ge b l lUn H kZ xza Fk l s js[ kh; xfr d s lH kh l w= i zkI r d j

ld r s gSA t S ls fd

v = u + at

r Fkk

v2= u2 + 2as

de Z H kh p kj i zd kj d s eku s x; s Fks

1111 ---- mR{ ksi . k ¼mR{ ksi . k ¼mR{ ksi . k ¼mR{ ksi . k ¼upward motion½½½½

2222 ---- vo{ ksi . k ¼vo{ ksi . k ¼vo{ ksi . k ¼vo{ ksi . k ¼downward motion½ ½ ½ ½

3 - vk d qvk d qvk d qvk d q ap u ap u ap u ap u ¼¼¼¼motion due to tensile stress½½½½

4444 ---- i zl k j. k ¼i zl k j. k ¼i zl k j. k ¼i zl k j. k ¼Shear freed motion½ ½ ½ ½

4444 ---- f LF kf r LF k ki dr k l a L dk j ¼ f LF kf r LF k ki dr k l a L dk j ¼ f LF kf r LF k ki dr k l a L dk j ¼ f LF kf r LF k ki dr k l a L dk j ¼Elastic Forces½½½½

oS' ksf" kd n'kZu esa elasticity d ks fL Fk fr LFkki d r k d gk x ; k gSA b l d k o. kZu U; k;

d Un y h d s , d 'y ksd e sa d qN b l i zd kj gS&

24

r ki e ku d h r jg B ksl o Lr qv ksa e sa fLF kfr LFk ki d r k d k H kh xq. k g ks r k gS r Fkk b l

xq. k d s d kj. k B ksl o Lr qv ksa esa] cy yxku s i j t ks fo: i . k gksr k gS m l ls] o kfi l v iu k

v kdkj i zkIr d j ysr h gSA

dj d koy h e sa elasticity ¼ i zR; kLFkr k½ d ks fu Eu i zd kj le> k; k x; k gS&

elastic forces B kslksa r Fkk v n `'; in kFkksZ e sa m Ri Uu gksr s gS r Fkk b l h d kj. k i n kFk Z

Vibrate ¼d fEir ½ gksr s gSA

5555 ---- x q: R o rF k k x q: R o kd " k Z. k&x q: R o rF k k x q: R o kd " k Z. k&x q: R o rF k k x q: R o kd " k Z. k&x q: R o rF k k x q: R o kd " k Z. k&

oS' ksf" kd n 'kZu d s vu qlkj xq: Ro gh B ksl rFkk n zO ; d k u hp s dh v ksj fxju s d k

d kjd gSA ; g vn `'; gS r Fkk i n kFkksZ ds fx j u s ls gh b ld k Hkku gksr k gSA r Fkk xq: R o

d k Kku B ksl ksa d h H khr jh v kUr fjd d . kksa d s eki i j fu H kZj gSA

^U; k; d Un yh* d s o. kZu ds v u ql kj xq:R o d s d kj. k B ksl ksa d k i zR; sd d. k

, d l eku : i ls u hp s fxjr k gSA v FkkZr , d c M+ h o Lr q r Fkk , d NksV h o Lr q , d le ku

: i l s xq: R o d s d kj .k u hp s fxjr h gSA

tc fd i f'p eh ns'kksa esa v jLr q d s fl } kUr l s o Lr qv ksa d s u hp s fxju s d h n j d k

o Lr qv ksa d s H kkj d s l e ku qi kr h e ku k t kr k Fkk v FkkZr H kkj h o Lr q t Yn h u hp s fxj sxh r Fkk

gY d h o Lr q n sj lsA lu ~ 1 5 9 0 e sa xSfyfy; ksa u s ; g fl) fd ; k fd lHkh o Lr q, xq: Ro d s

d kj. k , d le ku xfr ls u hp s fxjr h gS] e xj ; g fl } kUr i zkp hu H kkjr esa _ f"k d . k kn_ f"k d . k kn_ f"k d . k kn_ f"k d . k kn

d s o S'ksf" kd n 'kZu e sa ekSt wn FkkA

,d vU; 'yksd esa ; g c r k; k x;k gS fd i `Fo h d h lr g ij i zR; sd o Lr q i `Fo h

d s d sUn z d h v ksj t kr h gqb Z js[ kk d h fn ' kk esa fxjr h gSA

HkkL d jkp k; Z ¼1 1 3 4 b Z-½ d s v u ql kj i `Fo h esa v iu h vksj [ khap u s d h { ker k gS rFkk

, d H kkjh f i . M ¼xq: ½ t ks c zãekaM e sa mi fLFkr gS v iu s H kkj d s l e ku qi kr ¼xq: Ro½ e sa

i `Fo h d ks [k hp r k gSA

vFkkZr c zge kaM d h i zR; sd o Lr q ,d n wljs d ks v i u h v ksj [ khap r h gSA b l h i zd k j

[ kxks yh; fi . Mksa dh xfr foK ku r Fkk mud h lV hd x. ku k, a v u sd i zkp hu xzaFkksa e sa

mY y sf[ kr gSA

l an H kZ i qLr d &l an H kZ i qLr d &l an H kZ i qLr d &l an H kZ i qLr d &Physic in Ancient India (A Mathematical Analysis of Vaisheshika

Philosophy)- N.G. Dongre Pub., Wiley Eastern Limited (New Age Publishers

Limited)-1995.

25

AA →

lfn’k dks n’kkZuk

A→

Z

X

YO

Ax

Az

Ay

lfn’k dk fo;kstu

6666---- vfn’k vkSj lfn’k vfn’k vkSj lfn’k vfn’k vkSj lfn’k vfn’k vkSj lfn’k (Scalars and vectors)&&&&

¼1½ vfn’k vkSj lfn’k& ¼1½ vfn’k vkSj lfn’k& ¼1½ vfn’k vkSj lfn’k& ¼1½ vfn’k vkSj lfn’k& dqN HkkSfrd jkf’k;kWa tSls le; (time)] nzO;eku (mass)] Å"ek (heat)] rkiØe (temperature) vkfn ,slh jkf’k;kWa gSa ftudh vfHkO;fDr ds fy;s fn’kk ds Kku dh vko’;drk ugha gksrh ,oa fdlh ,d vad ek= ls gh budks iwjh rjg ls vfHkO;Dr fd;k tk ldrk gSA ,slh jkf’k;kWa vfn’k (scalars) dgykrh gSaA ysfdu dqN HkkSfrd jkf’k;kWa tSls foLFkkiu (displacement)] osx (velocity)] laosx (momentum)] cy (force) vkfn ,slh Hkh gksrh gSa ftudks fcuk fn’kk ds Kku ds iwjh rjg ls vfHkO;Dr ugha fd;k tk ldrk gSA ,slh jkf’k;ksa dks lfn’k (vectors) dgrs gSaA

lfn’k jkf’k dks ge ;k rks cksYM o.kZ (bold latters) ds :Ik esa ;k fQj o.kZ ds Åij ,d rhj (arrow) [khap dj fy[krs gSaA o.kZ lfn’k ds ifjek.k dks n’kkZus ds fy;s gksrk gS tcfd cksYMusl vFkok ml ij ,sjks dk fpUg mldh fn’kk ls lacaf/kr jgrk gSA lfn’k jkf’k dks xzkQh; rjhds ls vfHkO;Dr djus ds fy;s ge ,d funsZ’k ra= dk lgkjk ysrs gSaA blesa lfn’k jkf’k ds ijhek.k (magnitude) ds cjkcj ,d js[kk [khaprs gSa rFkk ftl fn’kk esa HkkSfrd jkf’k fØ;k’khy gksrh gS ml fn’kk dh vksj js[kk ds fljs ij ,d rhj dk fpUg cuk nsrs gSaA bls ^osDVj ;k lfn’k^ ls lacksf/kr djrs gSaA fdlh Hkh lfn’k ( )A

r dks ml

funsZ’k ra= (frame of reference) dh v{kksa ¼tSls x-, y-,z-½ ds lkis{k fo;ksftr dj ds mlds lfn’k ?kVd ( )zyx AA,A

rrr izkIr dj ldrs gSaA

vr% lfn’k ?kVdksa ds :Ik esa lfn’k ( )Ar

dks ge fuEukuqlkj fy[krs gSa%

zyx AAAArrrr

++= ..(A)

ifjHkk"kk ds vuqlkj izR;sd lfn’k ?kVd funsZ’k ra= (frame of reference) dh ,d v{k dh vksj jgrk gSA vxj ge funsZ’k ra= (frame of reference) dh v{kksa ¼tSls x-, y-, z-½ ds lkis{k bdkbZ lfn’k ¼bdkbZ lfn’k os gksrs gSa ftudk ijhek.k bdkbZ ek=d gksrk gS vkSj os fdlh fufnZ"V fn’kk esa gksrs gSa½ j,i ˆˆ vkSj k dks ifjHkkf"kr djsa rks ge

xAr

dks iAxˆ vkSj blh rjg yA

r rFkk zA

r dks Hkh fy[k ldrs gSaA vr% lfn’k ( )A

r

kAjAiAA zyxˆˆˆ ++=

r ... (B)

;gkWa zyx A,A,A vad gSaA bUgsa Ar

lfn’k ds ?kVd dgrs gSaA vxj ge x y- ry esa fLFkr

fdlh lfn’k ij fopkj djsa vkSj ekusa fd lfn’k ( )Ar

/kukRed x- v{k ls θ dks.k cukrk gSs ¼;kn jgs ;gkWa /kukRed x- v{k ls dks.k θ dks ekik tk jgk gSA _.kkRed x- v{k ls θ dks.k dk eryc /kukRed x- v{k ls ¼π−θ½ dks.k gksxk½ rks gesa blds fuEu nks ?kVd ( )yx A,A

feysaxs%

AsinθA AcosθA yx == ... (C)

26

A

A

ΒΒΒΒθ

ΒΒΒΒ

fp=&3% xq.ku izfØ;k

ΒΒΒΒ

A cosθ B

AΒΒΒΒcosθA

fp=&4% B dk izkstsD’ku A ij rFkk

A dk izkstsD’ku B ij

ik;Fkkxksjl cks/kk;u izes; ls lfn’k dk ijhek.k%

( ) ( )2y

2x AAA += ... (D)

rFkk] f=dks.kferh (Trigonometry) dh lgk;rk ls lfn’k dh x- v{k ds lkis{k fn’kk fuEu O;atd ls izkIr dh tk ldrh gS %

x

y

A

Atanθ = ;kfu]

x

y

A

Aθ

1tan−= ... (E)

(2) MkWV xq.kuMkWV xq.kuMkWV xq.kuMkWV xq.ku (Dot product) :

fdlh fudk; ij dksbZ cy vkjksfir djus ls Tkc og dqN nwjh ij foLFkkfir gks tkrk gS rks ge dgrs gSa fd geus dk;Z fd;kA dk;Z dh x.kuk ds fy;s gesa ekywe gksuk pkfg;s fd geus cy fdruk vkSj fdl fn’kk esa yxk;k rFkk gesa foLFkkiu fdruk vkSj fdl fn’kk esa feykA cy rFkk foLFkkiu nksuksa lfn’k gSa vkSj buesa xq.kk djus ls gesa dk;Z dk ijhek.k izkIr gksrk gSA ysfdu] dk;Z dks n’kkZus ds fy;s gesa fn’kk ugha crkuk iM+rhA dk;Z ,d vfn’k jkf’k gSA bl rjg ge ns[krs gSa fd ;gkWa nks lfn’kksa A

r rFkk B

resa xq.kk djus ls gesa vfn’k ds :Ik esa

ifj.kke (resultant) feyrk gSA bl xq.ku&izfØ;k dks vfHkO;Dr djus ds fy;s ge Ar

rFkk Br

ds chp MkWV ( )⋅ dk iz;ksx djrs gSaA vr% MkWV xq.ku nks lfn’kksa ds chp gksus okyk ,d ,slk xq.kk gS ftldk ifj.kke gesa ges’kk ,d vfn’k ds :Ik esa feyrk gSA ;gh dkj.k gS fd bls ge vfn’k xq.ku Hkh dgrs gSaA

vfn’k xq.ku dks izkIr djus ds fy;s ge nksuksa lfn’k dks bl rjg [khaprs gSa fd muds fljs ,d fcanq ij vk tk;saA ekuk fd bl le; buds chp esa θ dks.k gSA ¼;g dks.k lnSo 0 ls 180o ds chp esa jgrk gSA½ vc ge vxj fp= dks ns[ksa rks ik;saxs fd lfn’k B

r

dk lfn’k Ar

ij izkstsD’ku θcosB gS vkSj ;g Ar

ds lekarj gSA vfn’k xq.kk dks ge lfn’k A

r ds ijhek.k

vkSj lfn’k Br

ds Ar

ij izkstsD’ku θcosB ds xq.kuQy ds :Ik esa ifjHkkf"kr djrs gSaA

)A(BcosBA θ=⋅rr

… (A)

vfn’k xq.ku dks ge lfn’k Br

ds ijhek.k vkSj lfn’k Ar

ds Br

ij izkstsD’ku

θcosA ds xq.kuQy ds :Ik esa Hkh ifjHkkf"kr dj ldrs gSaaA vr%

( )θAcosBAB =⋅rr

mi;qZDr laca/kksa ls ge ns[krs gSa fd vfn’k xq.kk esa Ar

rFkk Br

vkil esa Øefofues; (commute) djrs gSaA ;kfu]

ABBArrrr

⋅=⋅ … (B)

Ar

rFkk Br

dks ?kVd ds :Ik esa fy[kus ij

zzyyxx BABABABA ++=⋅rr

… (C)

27

Thumb

FingersBA

Thumb

Fingers

BA

lfn’k xq.ku izfØ;k esa fn’kk dk laKku

A

ΒΒΒΒθ

n

A ΒΒΒΒ

fp=&5% lfn’k xq.ku izfØ;k

vxj ge j,i ˆˆ vkSj k tks fd x, y rFkk z fn’kk esa bdkbZ lfn’k gSa] ds fy;s fopkj djsa rks gesa fuEu mi;ksxh O;atd feyrs gSa%

1kkjjii =⋅=⋅=⋅ ˆˆˆˆˆˆ

0ikkjji =⋅=⋅=⋅ ˆˆˆˆˆˆ

(3) ØkWl xq.kuØkWl xq.kuØkWl xq.kuØkWl xq.ku (Cross product) :

tc fdlh fudk; ij dksbZ leku ijhek.k okys cy ,d lkFk ,d gh fcanq ij foijhr fn’kkvksa esa fØ;k’khy gksrs gSa rks og fudk; lkE;koLFkk esa iM+k jgrk gSA ysfdu tc ;s gh cy vyx&vyx fcanqvksa ij vkjksfir gksrs gSa rks fudk; fdlh v{k ds ifjr% ?kweus dk iz;kl djrk gSA ;g ?kqeuk fudk; ij vkjksfir cy vkSj muds chp fo|eku yEcor nwjh ij fuHkZj djrk gSA fudk; dk ?kqeuk nks izdkj ls gks ldrk gSA og ;k rks ?kM+h dh fn’kk esa gksrk gS vFkok mldh foijhr fn’kk esaA ?kqeus ds fy;s vko’;d HkkSfrd jkf’k cy vk?kw.kZ (torque) dgykrh gSA ;g ,d lfn’k jkf’k gSA vk?kw.kZ dh x.kuk ds fy;s gesa cy rFkk muds chp dh yEcor nwjh dk vkil esa xq.kk djrs gSaA bl rjg ge ns[krs gSa fd ;gkWa nks lfn’kksa esa xq.kk djus ij gesa lfn’k ds :Ik esa ifj.kke feyrk gSA bl xq.ku izfØ;k dks vfHkO;Dr djus ds fy;s ge A

r rFkk B

r ds chp ØkWl ( )× dk

iz;ksx djrs gSaA vr% ØkWl xq.ku nks lfn’kksa ds chp gksus okyk ,d ,slk xq.kk gS ftldk ijh.kke gesa ges’kk ,d lfn’k ds :Ik esa feyrk gSA ;gh dkj.k gS fd bls ge lfn’k xq.ku Hkh dgrs gSaA

Ar

rFkk Br

ds chpa lfn’k xq.ku dks izkIr djus ds fy;s ge nksuksa lfn’kksa dks bl rjg [khaprs gSa fd muds fljs ,d fcanq ij vk tk;saA ;s lfn’k ,d ry esa fLFkr gksaxsA ekuk fd bl le; buds chp esa θ dks.k gSA lfn’k xq.ku

BArr

× dks ge ,d lfn’k ds :Ik esa ifjHkkf"kr djrs gSa ftldk ijhek.k θsinAB gksrk gSA T;kWferh;&:Ik esa ;g eku lekarj prqHkqZt (parallelogram) ds ml {ks=Qy ds cjkcj gksrk gS ftldh Hkqtk,Wa A

r rFkk B

r gksrh gSaA BA

rr× dh fn’kk ml ry ds yEcor gksrh gS

ftlesa Ar

vkSj Br

mifLFkr jgrs gSaA bl ry ds yEcor nks fn’kk;sa gks ldrh gSaA vr% BArr

× dh fn’kk dks izkIr djus ds fy;s ge θ dks Ar

ls Br

dh vksj ¼nks laHkkfor NksVs dks.kksa esa ls vis{kkd`r NksVs dks.k dks cukrs gq,½ /kukRed ysrs gq, ekirs gSaA BA

rr× dh fn’kk

izkIr djus ds fy;s ge nkfgus gkFk ds LØw (right handed screw) dk iz;ksx dj ldrs gSaA Ar

ls Br

dh vksj ?kqekus ij nkfgus gkFk dk LØw ftl fn’kk esa vkxs c<+rk gS] og BArr

× dh fn’kk dgykrh gSA ;kn jgs fdtc ge B

rls A

r dh vksj LØw dks ?kqekrs gSa rks ;g ihNs vkrk

gSA vr% BArr

× vkSj ABrr

× dh fn’kk,Wa ,d nwljs ds foijhr gksrh gSaA

ABBArrrr

×−=× ... (A)

bl rjg ABBArrrr

×≠× gS ftlls gesa irk pyrk gS fd lfn’k xq.kk esa A

r rFkk B

r vkil esa Øe fofue;

(commute) ugha djrs gSaA

BArr

× dh fn’kk izkIr djus ds fy;s ge vius nkfgus gkFk dh maxfy;ksa dks eqDds dh 'kDy esa bl rjg j[ksa fd gekjk vaxwBk ckgj dh vksj fudyk jgsA maxfy;ksa ds fljs A

r ls

28

Br

dh vksj ?kqekus dh fn’kk vkSj vaxwBk BArr

× dh fn’kk crk;sxkA

ABsinθBA =×rr

n .. .(B)

;gkWa n ,d bdkbZ lfn’k gS ftldh fn’kk ml ry ds yEcor gksrh gS ftlesa Ar

vkSj Br

fLFkr gksrs gSaA

nksuksa lfn’kksa ds chp ds dks.k θ ds 0 vFkok 180o gksus ij BArr

× dk eku 'kwU; gksrk gSA ;kfu] nks lekarj vFkok foijhr lekarj lfn’kksa dk xq.kuQy ges’kk 'kwU; gksrk gSA ysfdu] tc ;s nksuks lfn’k vkil esa yEcor gksrs gSa rc gesa BA

rr× dk vf/kdre eku feyrk gSA

?kVd ds :Ik esa fy[kus ij

zyx

zyx

BBB

AAA

kji

BA

ˆˆˆ

=×rr

… (C)

vxj ge Ar

rFkk Br

dks x, y rFkk z fn’kk esa bdkbZ lfn’k ekusa rks gesa fuEu mi;ksxh O;atd feyrs gSa%

0kkjjii =×=×=× ˆˆˆˆˆˆ … (D)

rFkk]

jik,ikj,kji ˆˆˆˆˆˆˆˆˆ =×=×=× … (E)

Elementary formulation of differential and integral Calculus

Actually these are the calculus of infinitesimal. These are the tools to handle the physical quantities that vary with respect to some variable.

Here the lines, surfaces and volumes are divided into a large number of very small parts. Then one considers the interrelations in the limiting case when size of each subdivision goes to zero

Differential Calculus

Let us consider a physical quantity, say displacement, changes in regular way as the value of time ‘t’ changes. Case I: displacement is proportional to time x(t) = At

Let us consider displacements at two moments of time one at ‘t’ and other at ‘t+∆t’ x(t) = A t x(t+∆t) = A (t+∆t)

the difference in displacements i.e. the distance travelled in time ‘∆t’ is

A (t+∆t) – A t = A (∆t)

On dividing by ∆t, we get exactly A. This is what is known as differential coefficient of At with respect to ‘t’.

Case II: displacement is proportional to square of time x(t) = At2

29

Let us consider displacements at two moments of time one at ‘t’ and other at ‘t+∆t’ x(t) = A t2 x(t+∆t) = A (t+∆t)2

The difference in displacements i.e. the distance travelled in time ‘∆t’ is

A (t+∆t)2 – A t2 = 2A t (∆t) + (∆t)2

On dividing by ∆t, we get 2At + ∆t. when ∆t is made infinitesimal, it can be neglected and we get differential coefficient of At2 with respect to ‘t’ as 2At.

In general differential coefficient of any function f(x) with respect to ‘x’ or the rate of change of f(x) with respect to ‘x’ is defined as

h

xfhxfh

dx

xdf )()(0lim

)( −+→=

Double and higher derivatives

Double derivative: The rate of change of dx

xdf )(with respect to ‘x’ is

=dx

xdf

dx

d

dx

xfd )()(2

2

Triple derivative: The rate of change of 2

2 )(dx

xfdwith respect to ‘x’ is

= 2

2

3

3 )()(dx

xfd

dx

d

dx

xfd

Similarly the differential coefficient of any function x(t) with respect to ‘t’ or the rate of change of x(t) with respect to ‘t’ is defined as

t

txttxt

dt

dx

∆−∆+→∆= )()(

0lim

If the function is ‘displacement’, we call this as velocity. Further, the rate of change of velocity is acceleration. The use of calculus is of great advantage in evaluating the motion related parameters.

Second equation of motion

s = ut +1/2 at2

On differentiating ‘s’ with respect to time, we get velocity ‘v’.

atudt

dsv +==

On differentiating ‘v’, we get acceleration ‘a’

2

2

dt

sd

dt

dva ==

30

Integral calculus:

We have seen above that the differential calculus considers relation between parts of geometrical figures, when the parts become infinitesimal.

The task of integral calculus s entirely opposite. It is to produce geometrical figures of final size by integration of infinitesimal parts.

∫∫ == xdxdtdt

dx

The symbol used in front of the integrand is nothing but an elongated ‘s’ standing for the word ‘sum’.

The integration of acceleration with respect to time means the result will be average velocity during the given time.

∫∫ === atadtdtdt

xd

dt

dx2

2

= v

The integration of velocity with respect to time means the result will be total displacement during the given time.

∫ ∫∫ ===2

2taatdtvdtdt

dt

dx= s

If v is constant,

∫ == svtvdt

The area under the velocity curve gives total displacement during the given time.

The area under the acceleration curve gives average change in velocity during the given time.

Definite integrals:

If the limits of integration are provided, then we write the limits over ingral symbol. The integration of (1) with respect to 'x' within limits x=0 to x=9 is written as and the value after integration is obtained as

9099

0

9

0

=−==∫=

=

xdxx

x

Let the integration of ( )xf with respect to 'x' is g(x). This integration within limits x=0 to x=9 is written as and the value after integration is obtained as

( ) ( ) ( ) ( )099

0

9

0

ggxgdxxfx

x

−==∫=

=

The integration of sinx within limits x = 0 to x = 2π is

( ) 00cos2coscossin2

0

2

0

=−−=−=∫ πππ

xxdx

31

Use of calculus

In order to use integral calculus, we need to divide the given system into infinitesimal suitable parts. After this we are to find the effect of our interest due to one such infinitesimal part. Then the effects due to all parts are integrated.

Line is divided into segments Plane is divided into strips Disc and shells are divided into rings Sphere is divided into shells

Generalization: The above elementary formulation based on calculum can be extended to two and three dimensions.

f}in izes;f}in izes;f}in izes;f}in izes;

HkkSfrdh esa lfUudVuksa dk cgqr egRo gSA lfUudVuksa dk iz;ksx djrs le; gesa O;atd esa iz;qDr izR;sd jkf'k;ksa ds ifj.kkeksa ij fopkj djuk gksrk gSA vxj gesa fdlh O;atd esa ,d jkf'k 1000 gS rFkk nwljh 1 gS rks ge 1000 dh rqyuk esa 1 dks ux.; eku dj NksM+ ldrs gSaA ysfdu 10 dh rqyuk esa 1 dks ux.; ugha ekuk tk ldrkA ekukfd gekjs ikl ,d O;atd gS% (1+x)n

ekuk fd 1 dh rqyuk esa x cgqr NksVk gS vc vxj ge (1+x)n dk foLrkj djsa rks

( ) ( ) ( )( ).......

!3

21

!2

111 32 x

nnnx

nnnxn n −−+−++=+

vc pw¡fd x cgqr NksVk gS rc x2, x3 …… x dh rqyuk esa vR;Ur gh NksVs gksaxs vkSj mUgsa ux.; ekurs gq, NksM+k tk ldrk gSA vr% HkkSfrdh; egRo dks /;ku esa j[krs gq, ( )nx+1 dks ( )nx+1 fy[kk tk ldrk gSA ,slk djus ij ?kVuk ds v/;;u ij xq.kkRed n`f"V ls dksbZ izHkko ugha iM+rk gSA

40

bdkb Z& 2 xfr fo Kkubdkb Z& 2 xfr fo Kkubdkb Z& 2 xfr fo Kkubdkb Z& 2 xfr fo Kku

xzkQ i j x f r dks n ’kkZu k] ,d leku xfr dk xzkQh; fu :i. k] v ki sf{ kd osx] i z{ ksI; xfr] o`r h; xfrA

HkkSfrd h d h d { kk i < +ku k vkj aHk d ju s d s i woZ fo|kf F k;ksa d ks lfn ’k o v fn’k d k K ku n su k vko’; d gSA ;g l e> ku k t : j h g S fd d kSu lh jkf ’k lf n’k g S vk Sj d kSu lh v f n ’k vkSj D; ksa g S aA d Sls n ks ;k n ks ls vf/kd lfn’kks a d ks t ksM+k vF ko k ?kV k; k t krk gS\ lfn ’kksa d k s t k sM +u s ;k ?k V ku s ls i gys fd u&fd u c krksa d k /;ku j[ ku k vk o’;d g S\ f d l izd k j n ks lfn’kksa d k s t ksM+ ;k ?kV k d j i fj.kke h lfn’k d ks i zkIr fd ;k t k ld rk g S\ b lh i zd kj fd lh lfn ’k d ks fd lrjg n ks yE co r f n’kkvk sa esa fo ;ksft r H kh f d ;k t k ld rk gS\

b ld s c kn mud k fofHkU u izd k j d h x fr;ksa ls i fjf p r d j ku k vko’ ;d g SaA xf r;k¡ , d & foeh ; f }&foeh ; ;k f=&fo eh ; g ks ld r h g SA t Sls fd lh i z{ ksI; d ks m) kZ/kj fn 'kk esa mN kyu s i j feyu s o kyh x fr , d &foeh ; xfr d gy krh g SA m)kZ/ kj fn'kk ls vyx f d lh H kh vU ; f n'kk esa iz{ ksI; d h xfr;k¡ f}&fo e h; g ksrh g SA b lh rjg ,d ljy j s[kk esa xfr d j jg s fi .M ksa d s VDd j , d & foeh; ys fd u , d r y es a n ks fi .Mksa d s c h p g ks jg h V Dd j f}&foe h; g ksrh g SA

x fr d k v/ ;;u d j rs le ; g esa b l c kr d k /;ku j[ ku k g ksrk g S fd vx j t ksM+u s ;k ?kV kb Z t k u s o kyh j kf’ k ;k¡ lfn ’k g Sa r ks mUg sa v kSj fo;ksft r d j x-v{ k o kyh j kf ’k ;ksa d ks vyx t ksM + ;k ?kV k d j i fj.kkeh x f r i zkIr d jsaA x fr d k v/;;u d jr s le; g esa d qN parameters d k K ku g ksu k t : j h g ksrk g SA vr % b ud ks le> ku s es a d qN vf/ kd le ; n su k vR ;Ur v ko’ ;d g SA d qN egRoi w.kZ parameters b l i zd kj gSa%

1- fo LF kkiu ¼ lfn'k ½ ] d ks.kh ; fo LF kki u ¼ lfn'k ½ vkSj n wj h ¼vfn 'k ½ 2- o sx ¼ lfn 'k½ vkSj p ky ¼ vfn'k ½ 3- n z O; e ku ¼ vfn'k ½ 4- j s[ k h; laos x ¼ lfn ' k½ v kSj d ks.kh; la o sx ¼lfn 'k½ 5- c y ¼l fn 'k½ vkSj c y vk?k w.kZ ¼ lfn ' k½ 6- x frt Å t kZ ¼vfn 'k½] f LF kfrt Åt kZ ¼vfn 'k ½] d k ;Z ¼vfn 'k ½ ] ' kfDr ¼v fn'k ½ 7- n kc ¼vfn ' k½

b uesa ls d qN parameters d k eku le ; d s lkF k ifj ofrZr g ksrk gSA d b Z c kj i fj o rZu d h n j rF kk d b Z c kj d qy i fj or Zu d k ek u t ku u k vko’;d g ksrk g SA b ld s fy;s vod yu d SY d qyl Differential Calculus vkSj l e k d yu d SYd q yl Integration Calculus d h t : j r iM +rh g SA

1111 ---- x zkQ i j x fr dks n' kkZu k x zkQ i j x fr dks n' kkZu k x zkQ i j x fr dks n' kkZu k x zkQ i j x fr dks n' kkZu k

b ld s fy;s x zkQ d h x- v{k d h vksj le ; vkSj y- v{ k d h v ksj fo LF kki u] o sx ;k Ro j.k d ks ysrs g SaA fQ j mfpr b Zd kb Z d k p;u d j voyksd u lkfj.kh d s vu qlk j x zk Q i j fofHkU u f can qv ksa d ks vafd r d j Le wF k oØ c u krs gSaA b ld s c kn oØ d k v/;;u d j Lyks i vkfn i j fo pkj d j rs g q, fo'ys"k .k d j rs gSaA oØ d s fd lh fc an q ij Lyksi d k v F kZ gS m l fc an q ij [ kh ap h xb Z Li ' kZ j s[ kk } kj k x- v{ k d s lkF k c uu s o kys dks.k d k ekuA vx j l e ; d s lkF k oØ d s g j fc an q i j Lyksi d k eku fu ; r jg rk gS rks oØ ljy j s[kk d s :i esa jg rh gSA b l ls i rk pyrk g S fd y- v{ k d h vksj i znf'kZr Hkk Sfr d j kf'k ] ;F kk fo LF kki u] o sx

41

;k Ro j.k vk f n ;k rks le; d s lkF k fu ;r jg rk g S ;k l e ; d s lkF k le: i r jhd s ls c nyrk g SA

Time

Displacement/ velocity

mi ;ZqDr x zkQ i j ljy js[kk n'kkZrh g S fd fo LF kki u ;k o sx le ; d s lkF k le: i rj hd s ls c < + jg k g SA

Time

Displacement/ velocity

mi ;ZqDr x zkQ i j ljy j s[kk n 'kkZrh g S fd fo LF kkiu ;k o sx le; d s lkF k fu;r c u k jgr k g SA

Time

Displacement/ velocity

42

mi ;ZqDr x zk Q n'kkZrh g S f d fo LF kki u ;k o sx l e ; d s lkF k yx krkj c ny j g k g SA

Time

Displacement/ velocity

mi ;ZqDr x zk Q fo LF kki u ;k osx esa l e; d s lkF k vko rhZ ofj o rZu d ks n'kkZrk g SA

l ki s{ k h ; o s x %l ki s{ k h ; o s x %l ki s{ k h ; o s x %l ki s{ k h ; o s x %

c l es a c SBd j c l d s vU n j j[ kh lH k h p ht ksa d h xf r d ks n sf[ k;s v kid s lki s{ k b ud k o sx ’kwU ; g SA ysfd u c kgj lM+d d s fdu kj s mx s isM + vkpd ks foi j hr fn 'kk esa p yr s g q, ut j vkr s g SaA 1v

� vkS j 2v

� o sx ls , d g h fn 'kk esa p y jg s n ks fi .M ksa d k lki s{ kh; o sx ( )21 vv

�� − g ksrk g S t c fd f oi j hr fn'kk esa p y jg s n ks fi.M ksa d k lki s{kh; o sx ( )21 vv

�� + g ksrk g SA

l aj f{ kr Hkk Sfrd j k f' k ;k¡l aj f{ kr Hkk Sfrd j k f' k ;k¡l aj f{ kr Hkk Sfrd j k f' k ;k¡l aj f{ kr Hkk Sfrd j k f' k ;k¡

x fr d s v/ ;;u d s n kSj ku d qN HkkSfr d j kf'k ;k¡ lajf {kr jgr h g Sa% Åt kZ ] j s[kh; laosx ] d k s.k h; l ao sx A b ld k er yc g qv k ] Åt kZ ¼? k V uk d s i w oZ½ = Åt kZ ¼? k V uk d s c kn½ j s[ kh; lao sx ¼? kV uk d s i woZ½ = j s[kh; lao sx ¼? k Vuk d s c k n½

d k s.k h; l ao sx ¼? kVuk d s i woZ½ = d ks.kh; l aosx ¼? kVu k d s c k n½

fu n Zs 'k r a=fu n Zs 'k r a=fu n Zs 'k r a=fu n Zs 'k r a=

x fr d k v/ ; ;u d j rs le; gesa ,d lanHk Z rd d h vko' ;d rk g ksrh gSA fc u k lanH k Z ra= d k mY ys[ k fd ;s c kr ug ha c u rh g S v kSj u gh d qN le> es a vkr k g SA lanH kZ ra= i z;ksx 'kk yk Hk h g ks ld r k g SA i z ;ksx 'kk yk esa f LF kj jg rs g q, g e fd lh ?kVu k d k o.kZu d j ld r s g SaA , slk r a= Hkh g ks ld r k g S t ks fLF kj r a= d s lki s{k lek u o sx ls x freku gksA fLF k j vkSj leku o sx ls x fr eku ra= t M +Ro h ; ra= d gyk rs gSaA ysfd u f LF kj ra= d s lkis{ k R of jr ;k ?k w.kZu ra= Hkh g ks ld rs g SaA ;s vt M +Ro h; ra= d gykrs g SaA vt M +Ro h; fun Zs'k r a=ksa es a Nn ~e c yksa d h l `f" V gksr h g SA c kg ~; es a > wyks a esa c SBd j b Ug sa eg lwl fd ; k t k ld rk gSA c kg j [ kM +s jgd j ¼t M+R o h; ra= ½ v kS j Rofj r fy¶ V ¼vt M+Ro h; ra=½ es a [ kM +s jgd j c yksa esa O;og kj esa var j d ks eg lwl d j fun Zs'k ra =ksa d s e gRo d ks l e > k t k ld r k g SA

43

i `F o h lw;Z d s lki s{ k ?k w.kZu vo LF kk esa g S vr% Lo kH kkfo d gh ;g v t M+Ro h; fun Zs 'k ra = g SA ysf d u i ` F o h d ks t M +Ro h; fun Zs' k ra = eku k t k ld rk g S D ;ksafd i`F o h d h xfr d s d kj.k ft u Nn ~e c yksa d k t U e g ksrk gS mud s eku vR ;U r ux .; gksr s g SaA lanH kZ ra= d k p qu ko H kh v R;U r vko' ; d g ksrk g S f d l i zd kj d s lanH kZ ra = d k p qu ko fd ; k t k ; ;g b l c kr i j fu HkZj d jr k gS fd ?kV u k d h lef efr d Slh g S\ i z {ksI; d h x fr] VD d j] ljy vko rZ xf r vkfn d s f y;s d kfV Zf'k ;u fun sZ'k r a= mi ; qDr g S ysfd u ?kw. k h Z x fr;ksa d s fy , / kzqo h; fun sZ'k ra = d k p ;u x f. kr h; n `f"V ls ykHk i zn gksr k g SA

fd lh ry es a fLF kfr l fn'k ;k fo LF kki u R�

X

Y

θ

R

x = R cosθ

y = R sinθ

b ls fo;ksft r d ju s i j ge sa b ld s fLF kfr lfn' k ;k f o LF kki u d s n ks ?kV d f e yrs gSaA

x ?kVd % θcosR

y ?kV d % θsinR

b ld k eryc ;g g qv k fd fd lh Hkh lf n'k d ks g e vi u h lqfo/kk u qlkj fo; ksft r d j ld r s g SaA fo;k sf t r d ju s d s fy, gesa , d fun Zs'k ra= d h vko ';d r k g ksrh g SA lk ekU ; r% fd lh Hkh ra= esa ge { kSfrt fn'kk esa x ?kV d vkSj m) kZ/kj fn'kk esa y ?kVd ysrs g SaA r y d s vfoyEc d h f n'kk esa z ?k V d ysrs g SaA ;gk¡ /; ku n su s ;ksX; c kr ;g g S fd fd lh Hkh rd i j n k s fn'kkvksa esa v fHky Ec [kh ap k t k ld rk gSA , d vksj c kgj d h vksj rF kk n wlj k van j d h vksj vc lo k y mB rk g S fd fd l fn ' kk esa g e z v{ k eku sax sA b ld s fy;s H kkSfrd h esa t ks i jEi j k g S mld s vu qlkj n kfgu s g kF k d s LØ w d ks yhft ;sA b ls l h/kk i d M + x ls y fn 'kk d h vksj ?kqe ku s i j ; g ft l f n'kk esa vkx s c< +rk gS og x fn'kk g ksrh g SA

b l rjg fd lh Hk h lfn 'k A�

d s rhu ? kVd f eyrs g SaA b ls g e fu E u ku qlkj fy[ k ld rs gSa%

zyx AAAA����

++=

;g k¡ xA�

, d , slk lfn' k g S t ks flQ Z x fn'kk esa g S

yA�

,d , slk lf n'k g S t ks fl QZ y fn'kk esa g S

44

zA�

,d , slk lf n'k g S t ks fl QZ z f n'kk esa gS

vc ge xA�

] yA�

vkSj zA�

d ks F kksM +k csg rj rj hd s ls fy[ k ld rs g Sa f t lls ;g

x f.krh; n ` f" V ls T;kn k mi ;ksxh c u ld sA bld s fy;s g esa x, y v kSj z fn' kkvksa esa b d kb Z lfn'kksa d ks i f jHkkf"kr d j rs g SaA

x fn'kk es a bd kbZ lfn'k og g S ft ld k i fj.kke , d g SA b ls z� vF ko k x fp Ug ls i znf'kZr d j rs g SaA ;k fu z� , d ,slk lfn'k g S ft ld k i fj.kke , d g S o fn 'kk x v{ k d h

vksjA vx j ge xA�

lfn 'k ft ld k i fj.kk e xx AA =�

g S d ks z� d s :i esa f y[ku k p kgas rks ;g

g ksx k% iAA xxˆ=

�

b lh rjg y vkSj z fn'k k esa

jAA yyˆ=

�

kAA zzˆ=

�

vc ge l e> ld rs g Sa f d x-y ry esa f LF kr lfn'k d s ek= n ks ?kVd ;kfu x vkSj y ;kfu f}fo eh ; xfr ;ksa d s v/;;u d s n kSj ku i z;qDr lfn 'kksa d ks n ks ?kVd ksa esa f o;ksft r d ju k vko' ;d g SA ,d foeh; x fr esa ek = , d g h ?kV d g ksrk g SA b ld h fn'kk vko' ; /ku kRed ;k _.kkR ed gksr k g SA b ld h fn'kk v o'; /ku kR ed ;k _.k kR ed g ks ld rh gSA

xfr dk v / ;;u xfr dk v / ;;u xfr dk v / ;;u xfr dk v / ;;u

, d foeh ; ; kfu m) kZ/kj xfr bl esa ;k n j[ku k gk sx k fd ge kj h xfr y fn 'kk esa g h g ks jg h g SA i`F o h d k x q: Roh; Ro j.k -y f n'kk esa ln S o lfØ ; jg rk g SA t c Hkh fd lh i z{ ksI; d k s Åi j d h vksj Q sad u s ij mld h x fr d e g ksrh g S D;ksafd i `F o h d s x q: Ro kd"kZ.k d s d kj.k m ld k o sx d e g ksu s yx rk gS t Sls g h bld k o sx 'kwU ; g ksrk g S ; g i `F o h d h vksj ykSV u s yx rk g SA , slk H kh g ks ld rk g S fd v kjafH kd o sx b ru k vf/k d g ks fd ;g i`F o h d s x q: Ro kd "kZ.k { ks= ls c kgj p yk t k ;A f t l U ;wu r e vkj afH kd o sx ls ;g g ksr k g S mls i yk; u o sx d grs g SaA b ld s v /;;u d s f y;s x fr d s lehd j.k d k leqf pr K ku vko';d g SA b lhfy ;s t :j h g S fd U ;wVu ds x fr laca/ kh fu; ek sa ls c Ppksa d ks i gys voxr d j k fn ;k t k ;sA x fr d s lehd j.k g Sa

;g k¡ Å) kZ/kj fn'kk d h vksj x fr d j jgs fi .M d s f y;s

v = u -g t

h = ut - ½ gt2

v2 = u2 - 2gh rF kk] u hp s d h vksj fx j jgs

v = u +g t h = ut + ½ gt2 v2 = u2 + 2gh

fd lh Hkh fi . M d h f LF kfrt Åt kZ MgR gksrh g SA

45

221 mvMgR =

R

GMgRv

22 ==∴

i z { ksI ; r y e sa i z { ksI ; r y e sa i z { ksI ; r y e sa i z { ksI ; r y e sa

;g k¡ g esa y- ¼ Å) kZ/kj½ d s lkF k g h x- ¼ { kSfrt ½ v{ k d h vksj g ksu s ok yh x f r i j H kh fop kj d ju k g ksrk g SA

eku k fd i z{ ksI; d k s fn;s t ku s ok yk o sx { kSfrt d s lkF k θ d ks.k cu krk g SA osx lfn' k j kf'k gS vr% b ls x vk Sj y fn'kk esa fo; ksft r d j n ks ?kVd ksa d s :i esa i zkIr fd ;k t k ld r k g SA vc y- fn'kk esa x fr d k v/; ;u B hd m lh r jg fd ;k t k ld rk g S ft l rjg ,d foe h; x fr d s n kSj ku fd;k t krk gSA c l ;g k¡ o sx d k eku vsinθ ysu k g ksx k rF kk Ro j.k d ks /; ku esa j[ ku k gksx k ysfd u x- fn'kk esa i z{ksI; d k v/;;u d j rs le; gesa fop k j d ju k g ksx k fd b l fn'kk esa d ksb Z Hkh Ro j. k lfØ ; ug ha g SaA ;k fu fd i zs{kI; d x fr d s n kSj ku x- fn 'k k e sa m ld kk o sx ;kfu vcosθ vi fj ofr Zr jg rk g S t c fd y- fn'kk esa , slk ug ha g ksrkA

i z{ksI; d s v/ ;;u d s n kSjk u , d c kr d k laK ku j[ ku k t : j h gSA og ;g fd b ld h x fr d s n kSj ku Åt kZ lajf{ kr jgrh g SA vkj aHk esa l kj h Åt kZ x frt g ksrh g SA i z{ ks I; u s Å) kZ/kj x eu d s n kSj k u bles a x fr t Åt kZ d e g ksrh t krh g S v kSj fLF kf rt Åt kZ c < +rh t krh g SA fQ j t c b ld k o sx 'kwU; g ksrk g S rks le Lr Åt kZ fLF kfr t Å t kZ esa cn y t krh g S vkSj x frt Åt kZ 'kwU; gks t krh g SA fQj t c ; g i z{ksI; u hp s vk u s yx rk gS rc xfr t Åt kZ i qu% c < +u s yx rh gS v kSj fLF kfr t Åt kZ ?k V u s yx rh g SA

d `fre mi x zgk sa d ks j kd sV d h lg k ;rk ls d { kk esa LF kkf i r fd ;k t k rk gSA

, d i z{ ksI; d ks i `F oh lrjg ls c gqr Å¡p kb Z i j i g q¡p k d j v xj m ls c g qr vf/kd { kSfrt o sx n s fn ;k t k; rks og i z{ksI; i `F o h d s mi x zg d s : i esa d k;Z d js y x sxkA

The basic physics involved in satellites (science of Satellites)

Basic formulas

1. r

mv

r

GMm 2

2=

r

GMv =∴

2. GM

r

v

rT

3

22 ππ ==

3. 2R

GMmmg = 2gRGM =∴

4. When satellite is very near to earth, replace r (distance of the satellite from the centre of earth) by R (the radius of earth). So, for satellites orbiting in low-lying orbits, we have

gRv =

And, g

RT =

46

5. Kinetic energy, Potential energy and total energy of satellite

Potential energy V(r) = -GM/r Kinetic energy K(r) = ½ Mv2 = -GM/2r (from 1) Total energy E(r) = -GM/2r = -K(r)

6.. Escape velocity:

r

GMmmv =2

2

1

Escape velocity from the surface of earth skmgRv earthescape /2.112, ==

j kd sV d k fl )kar%j kd sV d k fl )kar%j kd sV d k fl )kar%j kd sV d k fl )kar%

j kWd sV js[kh ; lao sx d s laj{ k.k d s fu; e d s vu qlkj dk;Z d jr k g SA jkWd sV esa t c b Za/ku d k ngu g ksrk gS rc ml esa ls vR ;f/kd o sx ls x Sl , d t sV&u ksnd ¼jet-propellant½ d s : Ik esa v iu s u hp s o kys Hkkx d h vksj ls c kg j fud yrh g S f t lls j kd sV d ks , d vko sx (impulse) f eyrk g SA b l vko sx (impulse) d s d kj.k j kWd sV tsV &u ksnd d s foi j hr fn’kk esa x fr d ju s yx rk g SA L ej.kh; gS f d x Sl d s fu d yu s ls j kWd sV d k n zO; eku yxkr kj d e g ksrk t kr k g S ft lls m ld k o sx yxkr k j c< +rk jg rk g S t c rd fd mld k bZa/ ku l ekIr ug ha g ks t krk A j s[ kh; lao sx laj{k. k d s fu ; e d ks /;k u esa j[k d j ge d g ld rs gSa fd ft l lao sx ls x Sl c kg j fud yrh gS m lh lao s x ls j kWd sV Åi j mB rk g SA

eku k fd f d lh le; ^t^ i j fd lh fun sZ'k ra= (i z;ksx 'kkyk ra=) d s lki s{ k ^M^ n zO; ek u o kys jkWd sV d k o sx V

� g SA ^t sV^ d s : Ik esa fu"d k flr xSlksa d s d kj. k ; fn j kWd sV d s n zO; eku

d s d e g ksu s d h n j dt

dMg S rF kk fu"d kflr x Slksa d k j kWd sV d s lki s{k osx v

� gks r ks

i z;ksx 'kk yk ra = esa x S lksa d k o sx ( )vV�� − rFkk x Slksa d ks izfr lsd .M f eyu s o kyk lao sx

dt

dM ( )vV�� − gksx kA ;g j kf’k g h j kWd sV i j yx u s o kys m l c y d s c j kcj g S t ks m ls Åi j

mBu s d s fy;s vko ’;d x fr i zn ku d j rh g SA vr%

=dt

)Vd(M�

dt

dM ( )vV��

− … (A)

;gk ¡ geu s xSlksa d s fu "d klu d s QyLo: Ik j kWd sV i j yx u s o kys c y d h r qyu k esa j kd sV d s Hkk j d ks mi s{ k.kh ; eku fy;k g SA vc ] mi ;qZD r lehd j.k d ks vod fy r d ju s ij]

dtdM

vdt

dMV

dtdM

VdtVd

M���

�

−=+

vF ko k] dt

dMv

dtVd

M�

�

−=

vF ko k] dMvVMd��

−= … (B)

mi ;qZDr l eh d j.k (B) d k s l ekd fyr d ju s i j

47

KlnMvV +−=��

… (C)

;gk ¡ K l ek d yu d k f u;rkad g S f t ld k eku vkj afH kd 'krksZa d s laK ku ls izkIr fd ;k t k l d rk gSA eku k fd j kWd sV d k vkj af Hkd n zO; eku 0M vk S j vkj af Hkd o sx oV

� gSA

vr % KlnMvV 00 +−=��

;kfu ] 00 lnMvVK��

+= gksx kA K d k eku l ehd j.k (C) esa j[ ku s i j g esa

M

MlnvVlnMvVlnMvV 0

000

������+=++−=

mi ;qZDr l eh d j.k ls Li "V g S fd i z;ksx 'kk yk fu n sZ’k ra= esa j kWd sV d s osx V�

ds vf/ kd eku d ks i zkIr d ju s d s fy;s (1) j kWd sV d s lki s{ k m l esa fu"d kflr g k su s o kyh x S lksa d k o sx vf/kd g ksu k p kfg;s rF kk (2) fd lh Hkh { k.k j kWd sV d k n zO;ek u mld s vk jafH kd n zO; ek u ls d e g ksu k p kfg;sA

52

bdkbZ &3 cy ,oa xfr ds fu;e

tM+Ro dh vo/kkj.kk] tM+Roh; ,oa vtM+Roh; funsZ'k ra= Nn~e cy] vfHkdsUnz Rpj.k] lery

,oa cafdr o`Rrh; ekxZ esa okgu dh xfrA

loZizFke tM+Ro dh vo/kkj.kk dks Li"V djus ds fy;s tM+Ro dk fu;e oLrq ij yxk;s loZizFke tM+Ro dh vo/kkj.kk dks Li"V djus ds fy;s tM+Ro dk fu;e oLrq ij yxk;s loZizFke tM+Ro dh vo/kkj.kk dks Li"V djus ds fy;s tM+Ro dk fu;e oLrq ij yxk;s loZizFke tM+Ro dh vo/kkj.kk dks Li"V djus ds fy;s tM+Ro dk fu;e oLrq ij yxk;s

x;s ckg~; cy ds fojks/k dx;s ckg~; cy ds fojks/k dx;s ckg~; cy ds fojks/k dx;s ckg~; cy ds fojks/k djus ds xq.k] oLrq dh eki ij vk/kkfjr dbZ O;kogkfjd mnkgj.k jus ds xq.k] oLrq dh eki ij vk/kkfjr dbZ O;kogkfjd mnkgj.k jus ds xq.k] oLrq dh eki ij vk/kkfjr dbZ O;kogkfjd mnkgj.k jus ds xq.k] oLrq dh eki ij vk/kkfjr dbZ O;kogkfjd mnkgj.k

izLrqr djsaA izLrqr djsaA izLrqr djsaA izLrqr djsaA

1111---- tMRo dh vo/kkj.kk %& tMRo dh vo/kkj.kk %& tMRo dh vo/kkj.kk %& tMRo dh vo/kkj.kk %& tM+Ro xq.k dh [kkst xSysfy;ksa us dh Fkh A nSfud thou esa ge ;g

ikrs gS fd oLrq;s vius&vki gh viuh fojke ;k xfr dh voLFkk dks ifjofrZr ugha dj ikrh

gSaA ;fn ge mldh bl voLFkk esa ifjorZu djuk pkgrs gSa] rks gesa ml ij cy yxkuk gksxkA

;fn fdlh Vscy ij dksbZ fdrkc j[kh gqbZ gS] rks mls gVkus ds fy;s gesa ,d cy dh

vko';drk gksxhA dejs esa Vscy dqlhZ vkfn j[ks gksus ij os rc rd vius LFkku ij j[ks jgsaxs

tc rd fd ge cy yxkdj mUgsa ugha gVk nsaA blh izdkj dksbZ oLrq ?k"kZ.k jfgr ry ij

xfr dj jgh gks rks rc rd xfr djrh jgsxh tc rd fd dksbZ cy yxkdj mls jksd u

fn;k tk;A ,d xsan dks tehu ij yq<+dk nsus ij og dqN nwj tkdj :d tk;sxh D;ksafd xsan

ij ?k"k.kZ cy rFkk gok dk ncko dk;Z djrk gS] ftlls og dqN nwj tkdj :d tkrh gSA

bu lc mnkgj.k ls ;g Kkr gksrk gS] fd ^^fdlh oLrq dk og xq.k ftlds dkj.k og

viuh fojkekoLFkk ;k xR;koLFkk esa Lor% gh ifjorZu djus esa vleFkZ gksrh gS] mldk tM+Ro

dgykrk gSA**

fdlh oLrq ds tM+Ro dh eki mlds nzO;eku ls dh tkrh gS A vFkkZr~ ftl oLrq dk

nzO;eku ftruk vf/kd gksxk mldk tM+Ro mruk gh vf/kd gksxk vr% mls xR;koLFkk esa ykus

ds fy;s vf/kd cy dh vko';drk gksxh A

vko';d funsZ'kvko';d funsZ'kvko';d funsZ'kvko';d funsZ'k tMRo dh vo/kkj.kk dks tM+Ro ds eki ds }kjk le>kosaA

U;wVu ds f}rh; fu;e ds lw= ls F = ma

∴ a = f/m ;fn cy F fu;r jgs rks

a α 1/m tM+Ro dks vklkuh ls le>k tk ldrk gS A

53

mnkgj.k & ,d dkj rFkk ,d gkFk Bsys dks leku cy ls /kdk;k tk; rks gkFk Bsys es vf/kd

Roj.k mRiUu gksxk D;ksafd mldk nzO;eku dkj ds nzO;eku dh rqyuk esa de gS A

tM+Ro ds izdkj %&tM+Ro ds izdkj %&tM+Ro ds izdkj %&tM+Ro ds izdkj %&

tM+Ro nks izdkj dk gksrk gS &

1- fojke dk tM+Ro 2- xfr dk tM+Ro

1- fojke dk tM+Ro %&fojke dk tM+Ro %&fojke dk tM+Ro %&fojke dk tM+Ro %& fdlh oLrq dk og xq.k ftles ckg~; cy dh vuqifLFkfr esa og

viuh fojkekoLFkk esa ifjorZu djus esa vleFkZ gksrh gsS] fojke dk tM+Ro dgykrk gS A

vFkok

;fn dksbZ oLrq fLFkj gS rks fLFkj jgsxh tc rd fd ml ij dksbZ ckg; cy ugha

yxk;k tk; A

mnkgj.k %&

1½ cl ds vpkud pyus ij mlesa cSBk O;fDr ihNs dh vksj >qd tkrk gS] D;ksafd

cl ds xfr'khy gksus ij 'kjhj ds uhps dk Hkkx xfr'khy gks tkrk gS] ijUrq

Åij dk Hkkx fojke ds tM+Ro ds dkj.k fLFkj gh jguk pkgrk gS A vr%

vpkud cl ds xfr'khy gksus ij O;fDr ihNs dh vksj >qd tkrk gS A

2½ isM+ dks fgykus ij mlesa yxs Qy uhps fxjus yxrs gSa A isM+ dks fgykus ij

mldh 'kk[kk;s xfr voLFkk esa vk tkrh gS tcfd Qy fojke ds tM+Ro ds

dkj.k fLFkj jguk pkgrs gSa] vr% os VwV dj uhps fxjus yxrs gSa A

2- xfr dk tM+Ro %&xfr dk tM+Ro %&xfr dk tM+Ro %&xfr dk tM+Ro %& fdlh oLrq dk og xq.k ftlds dkj.k og ckg~; cy dh vuqifLFkfr

esa viuh xR;koLFkk esa ifjorZu djus esa vleFkZ gksrh gS] xfr dk tM+Ro dgykrk gSA

vFkok

;fn dksbZ oLrq fdlh osx ls py jgh gS rks og mlh fn'kk esa mlh osx ls rc rd

pyrh jgsxh tc rd ml ij ckg~; cy u yxk;k tk; A oLrq ds bl xq.k dks xfr dk

tM+Ro dgrs gSA

mnkgj.k %&mnkgj.k %&mnkgj.k %&mnkgj.k %& cl ds vpkud :dus ij mles cSBk O;fDr vkxs dh vksj >qd tkrk gS] D;ksafd

cl ds xfr'khy gksus ij O;fDr Hkh mlh osx ls xfr'khy jgrk gS A tc vpkud cl :drh

54

gS rks mldk uhps dk fgLlk cl ds lkFk :d tkrk gS] ijUrq 'kjhj dk Åijh Hkkx tM+Ro ds

dkj.k xfr'khy cuk jgrk gS] vr% O;fDr vkxs dh vksj >qd tkrk gS A

¼¼¼¼

2222

½½½½

2222---- tM+Roh; rFkk vtM+Roh; funsZ'k ra= %& tM+Roh; rFkk vtM+Roh; funsZ'k ra= %& tM+Roh; rFkk vtM+Roh; funsZ'k ra= %& tM+Roh; rFkk vtM+Roh; funsZ'k ra= %&

funsZ'k Ýse nks izdkj ds gksrs gaS &

1- tM+Roh; Ýse os gksrs gSa] ftuesa U;wVu ds xfr ds fu;e ykxw gksrs gSaA tM+Roh; Ýse esa

;fn fdlh fi.M dk cy 'kwU; gS rks F = ma ls mldk Roj.k Hkh 'kwU; gksxkA

ekuk fd ,d funsZ'k Ýse esa dksbZ fi.M fojkekoLFkk esa gS A bldk funsZ'k Ýse

igys Ýse ds lkis{k ,d leku osx ls xfr'khy gS A vr% blds Ýse esa fLFkr izs{kd dks

fi.M ,d leku osx ls pyrk izrhr gksxk tcfd igys Ýse esa fLFkr izs{kd dks ogh

fi.M fojkekoLFkk esa izrhr gksxk A fdUrq nksuksa isz{kdks ds fy;s fi.M dk Roj.k 'kwU; gh

gksxk A vr% nksuksa Ýse tM+Roh; Ýse gksaxsA

bl izdkj os lHkh Ýse tks ,d nwljs ds lkis{k fojkekoLFkk esa gksrs gS] ;k

,d&nwljs ds lkis{k leku osx ls py jgs gksrs gSa tM+Roh; Ýse gksrs gSaA

i`Foh dh viuh /kqjh ij xfr rFkk lw;Z ds pkjks vksj dh o`Rrh; xfr esa mRiUu

vfHkdsUnz Roj.k ds eku dks de gksus ij bu nksuksa vfHkdsUnz Roj.kksa dh mis{kk djus ij

i`Foh dh lrg ij fLFkr izR;sd funsZ'k Ýse dks O;kogkfjd mn~ns';ksa ds fy;s tM+Roh;

Ýse ekuk tk ldrk gSaA

2- vtM+Roh; ÝsevtM+Roh; ÝsevtM+Roh; ÝsevtM+Roh; Ýse & vtM+Roh; Ýse os gksrs gSa] ftuesa U;wVu ds xfr ds fu;e ykxw ugha

gksrs gSaA

lHkh Rofjr rFkk ?kw.khZ funsZ'k Ýse vtM+Roh; Ýse gksrs gSaA eku yks ,d funsZ'k

Ýse s' tM+Roh; Ýse s ds lkis{k a0 Rofjr xfr dj jgk gSA vc ;fn ,d fi.M Qzse s

vko';d funsZ'kvko';d funsZ'kvko';d funsZ'kvko';d funsZ'k

fdlh d.k dh fLFkfr vFkok fdlh ?kVuk ds LFkku dks Bhd&Bhd iznf'kZr djus ds fy;s rhu ijLij yEcor v{kksa ds leqPp; dks funsZ'k ra= dgrs gSa ;k funsZ'k Ýse dgrs gSaA

55

ij fojkekokLFkk esa gS rks Qzse s ij fLFkr izs{kd mls fojkekoLFkk esa ns[ksxkA fdUrq Qzse

s' esa fLFkr izs{kd mls -a0 Roj.k ls Rofjr ns[ksxkA

fi.M ij cy F = - ma0 dk;Z djsxk tgka m fi.M dk nzO;eku gS A ijUrq

okLro esa fi.M ij dksbZ cy dk;Z ugha dj jgk gS A

bl izdkj dk cy tks okLro esa fi.M ij dk;Z ugha djrk] fdUrq funsZ'k Ýse ds

Rofjr gksus ij fi.M ij dk;Z'khy fn[kkbZ nsrk gS] Nn~e cy (fictitions force) dgykrk

gSA

mnkgj.k %& mnkgj.k %& mnkgj.k %& mnkgj.k %& tc dksbZ O;fDr dkj esa cSBdj fdlh rh{.k eksM+ ij eqM+rk gS rks dkj dh xfr

Rofjr gksrh gSA ml le; dkj esa cSBs O;fDr ij ,d cy dk;Z djrk gS] tks mls dkj ls

ckgj dh vksj ys tkus dk iz;kl djrk gSA ijUrq dkj ds ckgj [kM+s izs{kd dks fdlh izdkj dk

cy fn[kkbZ ugha nsrk gSA vr% dkj esa cSBs O;fDr ij yxus okyk cy vidsUnz cy gksrk gS]

bls gh Nsn~e cy Hkh dgk tkrk gSA

vtM+Roh; Ýse ds fi.M dh xfr dh O;k[;k U;wVu ds xfr ds fu;e ls ugha dh tk

ldrh gSA

3333---- vfHkdsUnz Roj.k %& vfHkdsUnz Roj.k %& vfHkdsUnz Roj.k %& vfHkdsUnz Roj.k %&

eku yks dksbZ d.k ,d leku pky ls o`rh; xfr dj jgk gS A bldh pky Hkys gh ,d

leku gksrh gS] ysfdu o`Rrkdkj iFk ij xfr gksus ls osx dh fn'kk yxkrkj cnyrh jgrh gS

D;ksafd bldk osx ifjofrZr gksrk gS] vr% ;g Rofjr xfr gSA pwafd osx dk ifjek.k fu;r

jgrk gS] osx dh fn'kk esa Roj.k dk ?kVd 'kwU; gksxkA vr% Roj.k lnSo osx ds yEcor dk;Z

djrk gSA pwafd o`rh; xfr esa osx lnSo Li'kZ & js[kk dh fn'kk esa dk;Z djrk gS] vr% Roj.k

dh fn'kk lnSo o`rh; xfr esa dsUnz dh vksj gksrh gSA blfy;s bls vfHkdsUnz Roj.k dgrs gSaA

56

v2 Q θ v1 B A

v θ

P v2 v1

P

fp= o`Rrh; xfr ,oa rkR{kf.kd lfn'k fu:i.k

fp= esa ,d o`Rrkdkj iFk fn[kk;k x;k gS] ml ij ,d d.k P ls xfr izkjEHk dj jgk

gS A o`rkdkj iFk dh f=T;k r gS A dqN le; t ds i'pkr~ d.k P ls Q ij igqprk gSA P ij

d.k dk izkjfEHkd osx v1 rFkk Q ij osx v2 gSS A tks fd o`Rrkdkj iFk dh Li'kZ js[kk dh

fn'kk esa gS A

ekuk fd ∠ POQ = θ

lfn'k vuqlkj PA = v1 , PB = v2 tks dze'k% P vkSj Q fcUnqij osx lfn'k dks

iznZf'kr djrs gSa] ftudk osxkUrj ∆v = v2 - v1 = AB gS A

∆PAB esa dks.k ¾ pki @ f=T;k

θ = ∆v / v - (1) [v 1 = v 2 = v ∴ pky ,d leku gS A ]

ijUrq dks.kh; osx ω = θ/ t ls

θ = ω.t -(2)

leh0 ¼1½ vkS ¼2½ ls

∆ v / v = ω.t ;k ∆ v / t = ω.v.

vr% lehdj.k ∆ v / t = ω.v. ls - (3)

jS[kh; Roj.k a = ∆ v / t = osx esa ifjorZu @ le; - (4)

vko';d funsZ'kvko';d funsZ'kvko';d funsZ'kvko';d funsZ'k o`Rrh; xfr djrs gq, d.k ds osx dh fn'kk fdl izdkj cnyrh gSaA le>krs gq, osxakrj }kjk mRié gksus okys Roj.k dks le>k;k tk;saA

57

leh0 (3) vkSj (4) ls

a = ω.v. - (5)

;fn v = ω.r. ls eku leh0 (5) es j[kus ij

a = ω.r. = ω 2 r

a = ω 2 r - (6)

;fn w = v/r ls d: leh0 (5) esa j[kus ij

a = v/r . v = v2 / r

a = v2 / r - (7)

leh0 (5), (6) ,oa (7) vfHkdsUnz Roj.k ds O;atd gS A

vfHkdsUnz Roj.k dh fn'kk %&vfHkdsUnz Roj.k dh fn'kk %&vfHkdsUnz Roj.k dh fn'kk %&vfHkdsUnz Roj.k dh fn'kk %& ∆ v

∆PAB es B A

∠PAB + ∠PBA + ∠BPA = 1800 V1 v2 v 1 = v 2 = v

∴ B = PA = v

vr% ∆PAB lef}ckgq ∆ gS A P

∴ ∠PBA = ∠PAB

2 ∠PAB + θ = 1800

2 ∠PAB = 180 - θ

∠PAB = 180 - θ/ 2

;fn θ 0 ij ∠PAB = 900

vr% osxkUrj ∆V, PA ,oa PB nksuksa ij yEc gSa A

vr% Roj.k] Li'kZ js[kk ¾ osx lfn'k ij yEc gksxk A

blfy;s Roj.k dh fn'kk o`Rrkdkj iFk ds dsUnz dh vksj gksxh A blfy;s bl vfHkdsUnz Roj.k

dgrs gSA

58

4444---- o`rh; lery lM+d ij dkj dh xfr %& o`rh; lery lM+d ij dkj dh xfr %& o`rh; lery lM+d ij dkj dh xfr %& o`rh; lery lM+d ij dkj dh xfr %&

eku fy;k fd nzO;eku dh ,d dkj fu;r osx ls f=T;k okys o`Rrkdkj lery iFk ij

xfr'khy gS A ml le; dkj ij fuEu cy dk;Z djrs gS &

R

F fs

mg

A½ dkj dk Hkkj mg uhps dh vksj tks fd vfHkyEc izfrfdz;k R }kjk larqfyr gksrk gS A

AA½ o`rkdkj iFk ij dkj dks ?kweus ds fy;s vko';d vfHkdsUnz cy F = mv2 / rdkj ds

Vk;j vkSj tehu ds e/; ?k"k.kZ cy }kjk izkIr gksrk gS A

vr% ?k"kZ.k cy F = µs.R

;k F = µs mg [∴R = mg] - (i)

o`rh; lery lM+d dks lqjf{kr ikj djus ds fy;s vfHkdsUnz cy dks ?k"k.kZ cy ds

cjkcj ;k mlls de gksuk pkfg;s A vFkkZr~

m ν2 / r < F

;k m v2 / r < µr R < µs mg

;k ν2 < µs rg

;k ν max = √µs rg - (2)

vko';d funZs'kvko';d funZs'kvko';d funZs'kvko';d funZs'k o`Rrdkj iFk ij ?kweus ds fy, vko';d vfHkdsUnz cy rFkk dkj ds Vk;j ,oa

i`Foh ds e/; yxus okys /k`"k.kZ cy dh vo/kkj.kk ls Nk=ksa dks voxr djkus ds Ik'pkr O;atd Kkr djsaA

59

lehdj.k ¼2½ vf/kdre osx dks iznf'kZr djrk gSA ftlesa dkj lqjf{kr o`Rrh; ekxZ dks

ikj dj ldsaA

5555---- cafdr lM+d ij dkj dh xfr & cafdr lM+d ij dkj dh xfr & cafdr lM+d ij dkj dh xfr & cafdr lM+d ij dkj dh xfr &

R fs

F

θ mg

m nzO;eku dh dkj <ky okyh o`rh; lM+d esa xfreku gSaA ftldh f=T;k r gS A

bldh xfr dks larqfyr djus ds fy;s bu ij Hkkj mg rFkk vfHkdsUnz cy mν2 /r dk;Z djrs

gSA budks larqfyr djus ds fy;s bu R rFkk F ds ?kVd cy fdz;k 'khy gksrs gS tks fd

fp=kuqlkj gSa A R ds ?kVd cy

i) RCos θ m/okZ/kj ?kVd ii) R Sin θ {kSfrr ?kVd

F ds ?kVd cy

i) F Cos θ {kSfrt ?kVd ii) F Sin θ m/okZ/kj ?kVd

vr%

RCos θ = mg + F Sin θ − (1)

R sin θ + F cos θ = mν2 / r - (2)

F = µ R - xfrd ?k"k.kZ xq.kkad gS A ¼3½¼3½¼3½¼3½

R Cos θ - F sin θ =mg - (4)

vko';d funZs'kvko';d funZs'kvko';d funZs'kvko';d funZs'k

ur lery ij xfr'khy dkj dks larqfyr djus okys ?kVd cy dh

vo/kkj.kk ls Nk=ksa dks voxr djkrs gq, o`Rrkdkj iFk ij dkj

dh xfr ds fy, O;atd Kkr djsaA

60

leh0 ¼2½ esa ¼4½ dk Hkkx nsus ij

mν2 /r = R sin θ + F Cos θ ¼R Cos θ ls mij uhps Hkkx nsus ij½

mg R cos θ + F sin θ

ν2 = R sinθ / R cos θ + F cosθ / R cosθ = tanθ + F/R

rg R cosθ / R cosθ - F sinθ / R cosθ 1-F/R tanθ

ν2 = tanθ + µ

rg 1 - µ tanθ

∴ ν2 = rg [tan θ + µ / 1 - µ tan µ] - (5)

lehdj.k ¼5½ ls Li"V gS fd dkj dh pky ν mlds nzO;eku ij fuHkj ugha djrh gS A

;fn Vk;j] tehu ds e/; ?k"k.kZ 0 gks rks leh0 ¼5½ ls

ν max = √ rg tan θ

61

tc ge ?kw.khZ xfr dh ckr djrs gSa rks ,d rks ge nwj ls [kM+s jgrs gq, ?kw.khZ xfr dks

ns[krs gSa vkSj nwljs ?kw.khZ xfr dj jgs nksyk pØ esa cSB dj D;k varj gS nksuksa fLFkfr;ksa esa\

o`Rrh; eksM+ ij Iysu lM+d vkSj vkSj Lyksi okyh lM+d ij py jgh dkj ij yxus

okys cyksa ij utj Mkfy;sA

;kn j[ksa fd dkj ij m)kZ/kj fn'kk esa dksbZ xfr ugha gS vr% bl fn'kk esa dqy cy

'kwU; gksrk gSA ysfdu dkj dks eqM+us ds fy;s vfHkdsUnz cy dh vko';drk gksrh gS rks mls

?k"kZ.k cy ds dkj.k feyrk gSA

o`Rrh; eksM+ ij Iysu lM+d ij py jgh dkj o`Rrh; eksM+ ij Iysu lM+d ij py jgh dkj o`Rrh; eksM+ ij Iysu lM+d ij py jgh dkj o`Rrh; eksM+ ij Iysu lM+d ij py jgh dkj mg dkj dk Hkkj N dkj ij izfrfdz;k f dkj ds ifg;ksa vkSj lM+d ds lEidZ ds dkj.k ?k"kZ.k cy

N

mg

f

o`Rrh; eksM+ ij Lyksi okyh lM+d ij py jgh dkjo`Rrh; eksM+ ij Lyksi okyh lM+d ij py jgh dkjo`Rrh; eksM+ ij Lyksi okyh lM+d ij py jgh dkjo`Rrh; eksM+ ij Lyksi okyh lM+d ij py jgh dkj

62

mg

N

f

mv2/R

θ

mv2/R dkj ij fdz;k'khy vfHkdsanz cy Lyksi okyh lM+d ds dkj.k N ;kfu dkj ij izfrfdz;k vkSj f ;kfu dkj ds ifg;ksa o lM+d

ds lEidZ ds dkj.k ?k"kZ.k cy dze'k% Å)kZ/kj vkSj {kSfrt fn'kkvksa esa ugha jg ikrs gSaA

N rFkk f lfn'k jkf'k;k¡ gSa vr% bUgsa Å)kZ/kj vkSj {kSfrt ?kVdksa esa ck¡Vuk gksrk gSA blds ckn

Å)kZ/kj vkSj {kSfrt fn'kkvksa esa dk;Zjr cyksa ij fopkj djrs gq, nks lehdj.kksa dks izkIr dj

leL;k ds lek/kku dh fn'kk esa vkxs c<+k tkrk gSA

73

o`Rrh; ekxZ ij xfr djrs n`<+ fi.M ds d.k

HkkSfrdh; v/;;u esa vdsyk d.k Hkh fudk; gS vkSj dbZ d.kksa ls fey dj cuus okyk fi.M Hkh fudk; gSA bldk eryc ;g gqvk fd ge gj oLrq ;k fi.M dks cgq&d.kh; fudk; dg ldrs gSaA cgq&d.kh; fudk; ds fofHkUu d.kksa ds e/; yxus okys cyksa dks vU;ksU; cy dgrs gSaA bl cy dk eku d.kksa ds chp dh nwjh ij fuHkZj djrk gSA vxj cká cy ds yxus ij fdlh fi.M ls lac) d.kksa ds chp dh nwjh (rij) ugha cnyrh gS rc ge ml fi.M dks n`<+ fi.M (rigid body) dgrs gSaA ;kfu] n`<+ fi.M ds fy;s vko’;d 'krZ fuEukuqlkj gksxh%

r ij = 0 … (A)

vxj cká cy vxj cká cy vxj cká cy vxj cká cy (external force) ds yxus ij ;g nwjh cnyrh gS rc ge ml fi.M ds yxus ij ;g nwjh cnyrh gS rc ge ml fi.M ds yxus ij ;g nwjh cnyrh gS rc ge ml fi.M ds yxus ij ;g nwjh cnyrh gS rc ge ml fi.M

dks fod`rh; fi.M dks fod`rh; fi.M dks fod`rh; fi.M dks fod`rh; fi.M (deformable body) dgrs gSaA izd`fr esa dksbZ Hkh fi.M iw.kZr% n`<+ ugha dgrs gSaA izd`fr esa dksbZ Hkh fi.M iw.kZr% n`<+ ugha dgrs gSaA izd`fr esa dksbZ Hkh fi.M iw.kZr% n`<+ ugha dgrs gSaA izd`fr esa dksbZ Hkh fi.M iw.kZr% n`<+ ugha gksrk gSA gkykafd O;ogkfjd n`f"V ls ge mu lHkh fi.Mksa dks n`<+ eku ldrs gSa ftuesa cká gksrk gSA gkykafd O;ogkfjd n`f"V ls ge mu lHkh fi.Mksa dks n`<+ eku ldrs gSa ftuesa cká gksrk gSA gkykafd O;ogkfjd n`f"V ls ge mu lHkh fi.Mksa dks n`<+ eku ldrs gSa ftuesa cká gksrk gSA gkykafd O;ogkfjd n`f"V ls ge mu lHkh fi.Mksa dks n`<+ eku ldrs gSa ftuesa cká cy dcy dcy dcy ds yxus ij d.kksa ds chp dh nwjh vis{kkd`r cgqr de cnyrh gS vkSj izk;ksfxd x.kuk s yxus ij d.kksa ds chp dh nwjh vis{kkd`r cgqr de cnyrh gS vkSj izk;ksfxd x.kuk s yxus ij d.kksa ds chp dh nwjh vis{kkd`r cgqr de cnyrh gS vkSj izk;ksfxd x.kuk s yxus ij d.kksa ds chp dh nwjh vis{kkd`r cgqr de cnyrh gS vkSj izk;ksfxd x.kuk ds nkSjku ftldh mis{kk dj ikuk laHko gksrk gSAds nkSjku ftldh mis{kk dj ikuk laHko gksrk gSAds nkSjku ftldh mis{kk dj ikuk laHko gksrk gSAds nkSjku ftldh mis{kk dj ikuk laHko gksrk gSA

gekjk lkekU; vuqHko crkrk gS fd tc Hkh dksbZ fi.M xfr djrk gS rc mlds lHkh d.k ,d&lkFk vkSj ,d gh rjhds ls xfr djrs gSA bldk eryc ;g gqvk fd ge izR;sd d.k dh xfr dks lEiw.kZ fi.M dh xfr Hkh eku ldrs gSaA ;g xfr fdlh Hkh izdkj dh (;Fkk] LFkkukUrjh;] nksyuh; vFkok ?kw.khZ) gks ldrh gSA /;ku nsus ;ksX; tks ckr gS og ;g gS fd ;gkWa lHkh d.k ,d lkFk vkSj ,d gh rjhds ls xfr djrs gSA vc] FkksM+k lksfp;sA tgkWa rd LFkkukUrjh; vkSj nksyuh; xfr;ksa dh ckr gS] gesa le>us esa dksbZ my>u ugha gksrhA ysfdu tc ge ?kw.khZ xfr dj jgs fi.M ds ckjs esa lksprs gSa rc ge ,d vyx izdkj dk fp= ikrs gSaA ?kw.khZ xfr dj jgs fi.M ds lHkh d.k o`Rrh; ekxZ (circular path) ij rks xfr djrs gSa ysfdu muds }kjk r; fd;s tkus okys o`Rrh; ekxksZa dh f=T;k,W vyx&vyx gksrh gSaA nwljs 'kCnksa esa] ?kw.khZ xfr dj jgs fi.M ds os d.k tks ?kw.kZu v{k ls vf/kd nwjh ij fLFkr gksrs gSa mUgsa ,d fuf'pr varjky esa ,d iwjk pDdj yxkus esa mu d.kksa dh rqyuk esa vf/kd nwjh r; djuk iM+rh gS tks ?kw.kZu v{k ls djhc fLFkr gksrs gSaA bldk eryc ;g gqvk fd fi.M dk tks d.k ?kw.kZu v{k ij fLFkr gksrk gS mls dqN Hkh nwjh r; ugha djuk iM+rh gSA bl rjg ge ns[krs gSa fd v{k ij fLFkr fi.M ds d.k fLFkj jgrs gaSA vc vxj ge dYiuk djsa fd gekjk fi.M fdlh v{k ls dqN nwjh cuk;s j[krs gq, pDdj yxk jgk gS rc ge ikrs gSa fd ml v{k ds lkis{k fi.M dk dksbZ Hkh d.k fLFkj ugha gSA ,sls esa fi.M dh ?kw.khZ xfr dks le>uk ,d leL;k gSA

nzO;ek u dsa nznzO;ek u dsa nznzO;ek u dsa nznzO;ek u dsa nz (Centre of mass) : mi;qZDr leL;k ls NqVdkjk ikus ds fy;s oSKkfudksa us ,d rjhdk lkspkA blds vuqlkj fi.M esa ge ,d ,sls fcanq dh dYiuk djrs gSa tks v{k ds ifjr% Bhd mlh izdkj ls xfr djrk gqvk izrhr gksrk gS ftl rjg ls fi.M ds lEiw.kZ nzO;eku okyk ,d d.kA bldk eryc ;g gqvk fd ge fi.M dh dYiuk ,d ,sls vdsys d.k ds :i esa djrs gSa ftldk nzO;eku fi.M ds nzO;eku ds cjkcj gksrk gSA fi.M esa fLFkr

74

m1

m2

r1

r2

rCM

f}&d.k ra=

m1 mi

r1

r2

rCM

ri

n&d.kh; ra=

,sls fcanq dks ge fi.M dk nzO;eku dsanz (Centre of mass) dgrs gSaA bl rjg bl dsanz ij oLrq dk lEiw.kZ nzO;eku dsafnzr gS] ,slk ge eku ldrs gSaA n`<+ fi.Mksa dh xfr dks le>us dh fn'kk esa ;g ,d egRoiw.kZ dne gSA vkb;s] igys ge nzO;eku dsanz dh /kkj.kk dks dqN mnkgj.kksa dh lgk;rk ls le>us dk iz;kl djrs gSaA ge vius vklikl fofHkUu vkdkj&izdkj dh oLrqvksa (tSls IysV] dkih] :yj vkfn) dks ns[krs gSaA dHkh&dHkh vkius bUgsa viuh maxyh ij cSysal djus dk iz;kl Hkh fd;k gksxkA oLrq dk og LFkku tgkWa vki cSysal djus esa lQy gq, gksaxsa] ogkWa gh ml oLrq dk nzO;eku dsanz (Centre of mass) fLFkr gksrk gSA fu;fer oLrqvksa ds fy;s nzO;eku dsanz dk irk yxkuk xf.krh; n`f"V ls vis{kkd`r vklku gksrk gSA mnkgj.k ds fy;s] xksykdkj IysV] fjax] Bksl ;k [kks[kys xksys ds fy;s muds dsanz] nzO;eku dsanz gksrs gSaA vk;rkdkj] oxkZdkj vkfn oLrqvksa ds fy;s muds fod.kksZa (diagonals) ds dkV fcanq] (point of intersection) nzO;eku dsanz gksrs gSaA ;gk¡ ;g Lej.kh; gS fd nzO;eku dsanz dk oLrq esa fLFkr gksuk t:jh ugha gSA ;g oLrq ds ckgj Hkh gks ldrk gS tSlk fd ge ,d fjax esa ns[krs gSaA nzO;eku dsanz ls gekjk rkRi;Z ek= bruk gS fd ogkWa ge oLrq ds lEiw.kZ nzO;eku ds gks ldus dh dYiuk djrs gSa rkfd xf.krh; n`f"V ls gesa lqfo/kk gks ldsA ,d vkSj /;ku nsus ;ksX; ckr gSA ,d yEcs rkj dks ge mlds e/; ls cSysal dj ldrs gSa ysfdu tc blh rkj dh fjax cuk nh tkrh gS rc bldk nzO;eku dsanz f[kld dj fjax ds dsanz esa pyk tkrk gSA ;kfu] Hkys gh oLrq dk nzO;eku u cnys] ysfdu ;fn bldk vkdkj cnyrk gS rks nzO;eku dsanz dh fLFkfr Hkh cny tkrh gSA vxj ge bl voyksdu ds fu"d"kZ dks foLrkj nsa rks ikrs gSa fd oLrq esa nzO;eku ds forj.k ij gh nzO;eku dsanz dh fLFkfr fuHkZj djrh gSA

vkb;s] vc bls ge ,d rkfdZd tkek igukus dk iz;kl djrs gSaA igys ge ,d ,sls lS)kafrd fudk; dh ckr djrs gSa ftlesa flQZ nks gh d.k gSaA blds ckn ge ml fudk; dh ckr djsaxsa ftlesa ^n^ d.k gksaxsA

ge tkurs gSa fd fdlh Hkh fudk; ds d.kksa dh fLFkfr dks n'kkZus ds fy;s ge ,d funsZ'k ra= dk mi;ksx djrs gSaA bl funsZ'k ra= esa d.kksa dks ge ^fLFkfr&lfn'k^ (position vector) ls vfHkO;Dr djrs gSaA

(B) f}&d.kh; fudk; ;k f}&d.k ra= f}&d.kh; fudk; ;k f}&d.k ra= f}&d.kh; fudk; ;k f}&d.k ra= f}&d.kh; fudk; ;k f}&d.k ra= (Two-particle system) % % % %

ekuk fd ;g fudk; m1 vkSj m2 nzO;eku okys d.kksa ls fey dj cuk gS ftlds ^fLFkfr lfn'k^ (position vectors) 1r

r vkSj 2rr gSaA vc

ifjHkk"kk ds vuqlkj nzO;eku dsanz og fcanq gksuk pkfg;s tgkWa fudk; dk lEiw.kZ nzO;eku ;kfu (m1 + m2) dsafnzr gksA ekuk fd bl fcanq dk funsZ'k dsanz ds lkis{k ^fLFkfr&lfn'k^ (position vector) cmr

r gSaA

vr% lkekU; rdZ ds vk/kkj ij fi.M ds ^nzO;eku vk?kw.kZ (nzO;eku vkSj funsZ'k dsanz ds lkis{k izkIr ^fLFkfr&lfn'k* dk xq.kuQy) ds fy;s ge dg ldrs gSa fd

( ) ( )2211cm21 rmrmrmmrrr

+=+ …(A)

( )( )21

2211cm mm

rmrmr

++

=rr

r …(B)

75

(C) n ^ d.kh; fudk; ;k cgq&d.k r a=d.kh; fudk; ;k cgq&d.k r a=d.kh; fudk; ;k cgq&d.k r a=d.kh; fudk; ;k cgq&d.k r a= (n-particle system) % % % %

ekuk fd M nzO;eku okyk ;g fudk; m1, m2, m3 … mn nzO;eku okys ^n^ d.kksa ls fey dj cuk gS ftlds ^fLFkfr lfn'k^ (position vectors) n321 r...r,r,r

rrrrgSaA vc ifjHkk"kk ds

vuqlkj nzO;eku&dsanz og fcanq gksuk pkfg;s tgkWa fudk; dk lEiw.kZ nzO;eku (m1 + m2) dsafnzr gksA ekuk fd bl fcanq dk ^fLFkfr&lfn'k^ (position vector) ( )cmr

r gSA mi;qZDr rdZ dks foLrkj nsrs gq, ge dg ldrs gSa fd ^n^ d.kh; fudk; d.kh; fudk; d.kh; fudk; d.kh; fudk; ds fy;s

( ) ( )nn332211cmn321 rm...rmrmrmrm...mmmrrrrr

++++=++++

( )( )n321

nn332211cm m...mmm

rm...rmrmrmr

++++++++

=rrrr

r … (C)

vFkok]

====

∑∑∑∑

∑∑∑∑

ii

iii

cm

m

rm

r

r

r … (D)

;fn ge nzO;eku dsanz ( )cmrr

ds funsZ'kkadksa dks X,Y,Z ekusa rks

=

∑

∑

ii

iii

m

xm

X ,

=

∑

∑

ii

iii

m

ym

Y vkSj

=

∑

∑

ii

iii

m

zm

Z gksaxsA

tgkWa] Mmi

i =

∑ rFkk i = 1,2,3…n gSA

Hkys gh fi.M NksVs&NksVs d.kksa ls fey dj cuk gks] ysfdu O;kogkfjd :Ik esa fi.M esa nzO;eku dk forj.k vlrr (discrete) izrhr ugha gksrkA vr% d.kksa dks igpku ikuk vkSj x.kuk ds fy;s mudk nzO;eku izkIr djuk laHko ugha gksrkA bl leL;k ls cpus ds fy;s ge xf.kr esa of.kZr lekdy dSydqyl dh rduhd dk iz;ksx dj nzO;eku&dsanz ds funsZ'kkad ds fy;s vko';d lw= izkIr djrs gSaA blds fy;s ge fi.M dks dm nzO;eku ds vR;ar NksVs&NksVs Hkkxksa esa foHkkftr djrs gSaA ekuk fd buesa ls fdlh ,d Hkkx dk ^fLFkfr&lfn'k^ (position vector) r

rgS rc funsZ'k dsanz ds lkis{k nzO;eku&dsanz dk fLFkfr

lfn'k gksxk%

( )

∫∫=

dm

dmrrcm

rr

… (E)

vxj ge gekjs funsZ'k&dsanz dks fi.M ds nzO;eku&dsanz ij fLFkr ekusa rks cmrr

dk eku 'kwU; gks tk;xk rFkk blds funsZ'kkad (0,0,0) gksaxsA ,slh fLFkfr esa nzO;eku&dsanz ds lkis{k fuEu 'krZ ges’kk ykxw gksxh%

( ) 0rm...rmrmrm nn332211 =++++rrrr

dSydqyl dh Hkk"kk esa

( ) 0dmr =∫r

… (F)

76

;gk¡ dm nzO;eku okys d.k dk nzO;eku vk?kw.kZ ( )dmrr

gS

;g vR;ar egRoiw.kZ ifj.kke gS tks ;g crkrk gS fd nzO;eku&dsanz ds ifjr% nzO;eku&vk?kw.kZ (moment of mass) dk eku ges’kk 'kwU; gksrk gSA

77

↑Z

m1

m2

m3m4

m5

r1

r2

r3

r4r5

mi ri

A

B

?kw.kZu djrk gqvk n`<+ fi.M

tM+Ro vk?kw.kZ tM+Ro vk?kw.kZ tM+Ro vk?kw.kZ tM+Ro vk?kw.kZ

cká cy dh mifLFkfr esa n`<+ fi.M ds lHkh d.k ;k rks lekUrj iFkksa ij vxzlj gksrs gSa vFkok cy&vk?kw.kZ dh mifLFkfr esa os lHkh o`Ùkkdkj ekxksZa ¼ftudk dsanz ?kw.kZu v{k ij fLFkr gksrk gS½ ij py jgs gksrs gSaA blh dks /;ku esa j[krs gq, ge dg ldrs gSa fd n`<+ fi.Mksa esa fuEu nks izdkj dh xfr;kWa mRiUu gks ldrh gSa%

(1) js[kh; xfr

(2) ?kw.khZ xfr

Lej.kh; gS fd js[kh; xfr ds fy;s cy dh vko';drk gksrh gS ftlls fi.M esa js[kh; foLFkkiu] js[kh; osx vkSj js[kh; Roj.k mRiUu gksrs gSa tcfd ?kw.khZ xfr ds fy;s cy&vk?kw.kZ dh t:jr gksrh gS ftlls fi.M esa dks.kh; foLFkkiu] dks.kh; osx vkSj dks.kh; Roj.k mRiUu gksrs gSaA

tM+Ro dh rjg tM+Ro vk?kw.kZ dk Hkh HkkSfrd egRo gSA tgkWa tM+Ro ds dkj.k fi.M viuh js[kh; xfr dk fojks/k djrk gS ogha tM+Ro vk?kw.kZ ds dkj.k fi.M viuh ?kw.kZu xfr dk fojks/k djrk gSA /;ku jgs fd fdlh fi.M dk tM+Ro vk?kw.kZ ftruk vf/kd gksrk gS] mls ?kqekus ds fy;s gesa mruk gh vf/kd vk?kw.kZ yxkuk iM+rk gSA vr% ge tM+Ro vk?kw.kZ dks ?kw.kZu xfr dk tM+Ro Hkh dg ldrs gSaA bl rjg ;g HkkSfrd jkf’k tM+Ro vk?kw.kZ fi.M dh viuh pfj=xr~ fo'ks"krk cu dj lkeus vkrh gSA

n`<+ f i.M dk tM+ Ro vk?kw.k Z %n`<+ f i.M dk tM+ Ro vk?kw.k Z %n`<+ f i.M dk tM+ Ro vk?kw.k Z %n`<+ f i.M dk tM+ Ro vk?kw.k Z % ge ,d n`<+ fi.M ysrs gSaA ekuk fd ?kw.kZu v{k ls bl fi.M ds d.kksa ...)...mm,m,(m i321 dh nwfj;kWa .......rr,r,r i321 gSaA ge tkurs

gSa fd mi nzO;eku okys d.k dk ?kw.kZu v{k ds lkis{k tM+Ro vk?kw.kZ

2ii rm gSA vr% lHkh d.kksa ds bl

rjg ls izkIr tM+Ro vk?kw.kks± dks tksM+us ij gesa ?kw.kZu v{k AB ds ifjr% iwjs fi.M dk tM+Ro vk?kw.kZ izkIr gks tkrk gSA

∑=

=n

1i

2ii rmI

;fn fi.M vfofPNUu (continuous) gS rc ge fi.M dks vR;ar NksVs&NksVs Hkkxksa dk cuk gqvk ekurs gq, lekdfyr dj ds tM+Ro vk?kw.kZ dks fuEukuqlkj izkIr djrs gSa%

∫=m

0

2dmrI

78

if jHkze.k f =T;kif jHkze.k f =T;kif jHkze.k f =T;kif jHkze.k f =T;k (Radius of gyration) : ftl rjg geus js[kh; xfr dj jgs fi.M esa nzO;eku dsanz dh ladYiuk dh Fkh mlh rjg fdlh ?kw.kZu v{k ds ifjr% xfr dj jgs fi.M ds fy;s Hkh ge v{k ls ^K^ nwjh ij fi.M esa ,d ,sls fcanq dh dYiuk djrs gSa tgkWa fi.M dk iwjk nzO;eku (M) ,df=r gksA ;g dYiuk xf.krh; n`f"V ls cgqr ykHknk;d gS D;ksafd vc ekuh xbZ ?kw.kZu v{k ds ifjr% lEiw.kZ fi.M dk tM+Ro vk?kw.kZ M K2 gks tk;sxkA bl rjg

∑=

==n

1i

2ii

2 rmMKI

vFkkZr] M

rmK

n

1i

2ii∑

==

bl ^K^ dks ge ,d ubZ HkkSfrd jkf'k ifjHkze.k f=T;k (radius of gyration) ds uke ls iqdkjrs gSaA ;g ?kw.kZu v{k ls og nwjh gS ftlds oxZ dk fi.M ds nzO;eku ls xq.kk djus ij gesa fi.M dk tM+Ro vk?kw.kZ izkIr gksrk gSA

tM+Ro vk?kw.kZ dks izkIr djus ds fy;s ge lekdy dSydqyl dk mi;ksx djrs gSaA fdlh Hkh oLrq dk tM+Ro vk?kw.kZ fudkyus ds fy;s ge mldh lefefr dks /;ku esa j[krs gq, mls cgqr gh vYi nzO;eku (dm) okys Hkkxksa ls cuk gqvk ekurs gSA fQj buesa ls fdlh ,d Hkkx ¼tks fd ?kw.kZu v{k ls x yEcor nwjh ij fLFkr gS½ dk tM+Ro vk?kw.kZ pkgs x;s ?kw.kZu v{k ds lkis{k (dm x2) fudky ysrs gSaA bl Hkkx ds fy;s izkIr tM+Ro vk?kw.kZ dks oLrq dh foLrkj

lhekvksa (ekuk fd a vkSj b) dks /;ku es j[krs gq, lekdfyr

∫b

a

2dmx dj ysrs gSa ftlls

gesa tM+Ro vk?kw.kZ dk okafNr eku fey tkrk gSA vko’;drkuqlkj lekarj vkSj ledksf.kd v{kksa ds izes; dk mi;ksx djrs gSaA vc ge bl rduhd dks Li"V :i ls le>us ds mís’; ls dqN lefer fi.Mksa ds tM+Ro vk?kw.kZ ds ekuksa dk ifjdyu djsaxsA

83

bdkbZ & 6 xq:Rokd"kZ.kbdkbZ & 6 xq:Rokd"kZ.kbdkbZ & 6 xq:Rokd"kZ.kbdkbZ & 6 xq:Rokd"kZ.k

xq:Ro Roj.k g ds eku esa ifjorZu] mixzg dh d{kh; pky] iyk;u osx Variation of 'g'

ge tkurs gSa fd

2d

mMGmg = ;kfu] 2d

GMg =

bl lw= esa i`Foh dh ?kw.kZu xfr dks ux.; ekurs gq, i`Foh dks tM+Roh; funsZ'k ra= ekuk x;k gSA

mi;qZDr lw= ls irk pyrk gS fd g dk eku nks izkpyksa ij fuHkZj djrk gSA ,d rks fopkjk/khu fi.M dh i`Foh dsanz ls nwjh vkSj nwljk i`Foh dk izHkkoh nzO;ekuA

iz;ksxksa ls feyh tkudkfj;ksa ds vuqlkj i`Foh le:i ugha gSA bldk ?kuRo vius dsanz ls lrg dh vksj c<+us ij fujarj ?kVrk tkrk gSA vr% g dk eku Hkh lrg ls dsanz dh vksj ¼tSls [knku esa½ tkrs gq, cnyrs gq, feyrk gSA lrg ls dsanz dh vksj tkrs le; nks ckrsa gksrh gSaA ,d rks i`Foh dk izHkkoh nzO;eku de gksrk tkrk gS ftlls g dk eku de gksrk tkrk gS vkSj nwljh fi.M dh i`Foh dsanz ls nwjh ?kVrh tkrh gS ftlls g dk eku c<+rk tkrk gSA

vc vxj ge i`Foh ds inkFkZ ds ?kuRo dks fu;r eku ysa rks g ds eku esa cnyko dks le>uk vklku gks tkrk gSA i`Foh ds inkFkZ ds ?kuRo dks fu;r ekuus dh fLFkfr esa d

xgjkbZ ij i`Foh dk izHkkoh nzO;eku 3

34

dπ gksxkA vr% 2d

GMg = dks ge fuEukuqlkj

vfHkO;Dr dj ldrs gSaA

g ∝2

3

3

4

d

d

πρ

;kfu] g ∝ d

bl rjg ge ns[krs gSa fd g dk eku lrg ls dsanz dh vksj ¼tSls [knku esa½ tkrs gq, ?kVrs gq, feyrk gSA

vc ge i`Foh dh lrg ls Å¡pkbZ ij p<+rs gq, g ds eku esa gksus okys ifjorZu ij fopkj djrs gSaA bl nkSjku fopkjk/khu fi.M dh i`Foh dsanz ls nwjh rks cnyrh gS ysfdu

i`Foh dk izHkkoh nzO;eku vifjofrZr jgrk gSA vr% 2d

GMg = dks ge fuEukuqlkj vfHkO;Dr

dj ldrs gSaA

g ∝ 2

1d

bl rjg ge ns[krs gSa fd g dk eku lrg ls Å¡pkbZ ij p<+rs gq, ¼tSls igkM+ ij½ tkrs gq, Hkh ?kVrs gq, feyrk gSA

84

pw¡fd iFoh piVh gS vr% g dk eku /kzqoksa ij vf/kd vkSj Hkwe/; js[kk ij ij de gksrk gSA

vc vxj ge i`Foh dh ?kw.kZu xfr dks ux.; ugha ekusa rks i`Foh ?kw.khZ ;kfu vtM+Roh; funsZ'k ra= dh rjg O;ogkj djsxhA vtM+Roh; funsZ'k ra= esa Nn~e cyksa dk mn; gksrk gSA ?kw.kZu ds dkj.k i`Foh lrg ij fLFkr fdlh Hkh LFkku ij ,d vidsanzh; cy

CFFrdk;Z djrk gSA ;g vidsanzh; cy λcosER f=T;k okys o`r dh ifjf/k ij fLFkr fi.M

ij ckgj dh vksj Hkwe/; js[kk okys ry ds lekukarj ry esa gksrk gS vkSj bldk eku λω cos2

ERm gksxkA vidsanzh; cy lfn'k jkf'k gS vr% bls nks Hkkxksa esa fo;ksftr fd;k tk ldrk gSA ,d rks fi.M dks i`Foh ds dsanz ls feykus okyh fn'kk esa rFkk nwljh blds yacor fn'kk esaA pw¡fd vidsanzh; cy dh fn'kk ckgj dh vksj gS vr% blds ,d ?kVd λcosCFF dh fn'kk Hkh ckgj dh vksj gksxhA bl rjg ;g cy i`Foh ds dsanz dh vksj yxus okys xq:Rokd"kZ.k dks de djsxkA

vr% i`Foh lrg ij fLFkr fdlh Hkh LFkku ij fi.M ij yxus okyk izHkkoh cy ;kfu Hkkj

λcos'CFFmgmg −=

tgk¡ FCF = λω cos2ERm gSA

110

MkIyj izHkko dh O;k[;kMkIyj izHkko dh O;k[;kMkIyj izHkko dh O;k[;kMkIyj izHkko dh O;k[;k /;ku nhft;s

IysVQkeZ ij [kM+s jg dj vkrh vkSj tkrh gqbZ Vªsu ls lqukbZ nsus okyh lhVh dks lqfu;sA igys lhVh dh vkokt ckjhd vkSj ckn esa eksVh gksrh lqukbZ iM+rh gSA /ofu ds lkFk tqM+h ;g ?kVuk gh MkIyj izHkko gSA

lzksr vkSj Jksrk ds chp fo|eku lkis{kh; xfr ¼pkgs og vdsys lzksr ds pyus ds dkj.k gks vFkok Jksrk ds pyus ds dkj.k gks vFkok nksuksa ds gh ds pyus ds dkj.k gkss½ ds dkj.k gh Jksrk rjax dh vko`fÙk esa vkHkklh ifjorZu eglwl djrk gSA

vxj nksuksa ,d leku osx ls ,d gh fn’kk esa vkxs ;k fiNs py jgs gSa ;k nksuksa fLFkj gksa rks Jksrk rjax dh vkofÙk esa vkHkklh ifjorZu eglwl ugha djrk gS D;ksafd ,slh fLFkfr esa nksuksa ds chp lkis{kh; xfr dk eku 'kwU; gksrk gSA vc bls xf.krh; Lo:i nsuk gSA

HkkSfrdh; n`f"V ls /ofu ,d izxkeh rjax gSA vkSj blhfy;s MkIyj izHkko rjax ls tqM+k izHkko gSA gesa ekywe gS fd vkokt dk ckjhd vkSj eksVk gksuk vko`fÙk esa cnyko dk ladsr gksrk gSA

rjax dh ifgpku fuEu HkkSfrd izkpyksa ls gksrh gS% 1- osx ¼v½ 2- vko`fÙk ¼n½ 3- rjaxnS/;Z ¼λ½

buds chp esa vkilh laca/k gksrk gS ftls rjax ds osx dh ifjHkk"kk ls

tkuk tk ldrk gSA fdlh rjax dks mldh rjaxnS/;Z ds cjkcj nwjh r; djus esa ,d vkorZdky dk le; yxrk gSA vkorZdky ek/;e ds fdlh ,d d.k dks ,d nksyu iwjk djus esa yxus okyk le; gksrk gSA vr% rjax dk osx

Tv

λ=

pw¡fd vkorZdky ,d nksyu esa yxus okyk le; gksrk gS vr% ,d lsd.M esa iwjs gksus okys nksyuksa dh la[;k ;kfu vko`fÙk gksxh%

Tn

1=

blrjg gesa osx] rjaxnS/;Z vkSj vko`fÙk esa fuEu laca/k feyrk gS%

111

nv λ=

λv

n =

vc gesa ;g ns[kuk gS fd dSls osx esa ifjorZu vko`fÙk esa cnyko ykrk gS\

bldk mÙkj ikus ds iwoZ gesa ek/;e dk Hkh /;ku j[kuk gksxkA ek/;e fLFkj Hkh gks ldrk gS vkSj pyk;eku HkhA igys ge fLFkj ek/;e ij fopkj djrs gSaA nksuksa fLFkj gksa%

,slh fLFkfr esa lzksr }kjk mRiUu rjax ds nks Jaxksa ds chp dh nwjh ¼λ½ vkSj Jksrk }kjk izkIr nks Jaxksa ds chp dh nwjh ¼λ′ ½ esa dksbZ varj ugha

'λλ == vT flQZ lzksr ¼vs osx ls½ pyk;eku gks%

tc lzksr ¼vs osx ls½ pyk;eku gksrk gS rks Jksrk dks ,slk vkHkkl gksrk gS ekuks lzksr }kjk mRiUu rjaxksa dh rjaxnS/;Z cny jgh gSA ,slk blfy;s gksrk gS D;ksafd lzksr }kjk mRiUu igyh rjax ds ckn og Lo;a vs T py pqdk gksrk gSA vr% vc lzksr nwljh rjax dks vius u;s LFkku ls mRiUu djrk gS ftlls Jksrk }kjk izkIr nks rjaxksa ds chp dk varjky cny tkrk gS vkSj og rjaxnS/;Z esa vkHkklh ifjorZu ds dkj.k vko`fÙk esa Hkh vkHkklh cnyko eglwl djus yxrk gSA vxj lzksr Jksrk ls nwj tkrk gS rks varjky c<+ tkrk gS ftlls vkHkklh vko`fÙk ?kV tkrh gSA ysfdu lzksr ds Jksrk dh vksj vkus ij ;g varjky ?kV tkrk gS ftlls vkHkklh vkofÙk c<+ tkrh gSA ekuk fd lzksr Jksrk dh vksj vk jgk gSA vr% lzksr }kjk mRiUu rjax dh rjaxnS/;Z% vT=λ Jksrk }kjk izkIr rjax dh rjaxnS/;Z% Tvv s )(' −=λ

Jksrk }kjk izkIr rjax dh vko`fÙk nvv

v

Tvv

vvn

ss )()(''

−=

−==

λ

flQZ Jksrk ¼vo osx ls½ pyk;eku gks

112

tc Jksrk ¼vo osx ls½ pyk;eku gksrk gS rks Jksrk dks ,slk vkHkkl gksrk

gS ekuks lzksr }kjk mRiUu rjaxksa dk osx cny jgk gSA ,slk blfy;s gksrk gS

D;ksafd fLFkj lzksr }kjk rks ,d gh LFkku ls rjax mRiUu dh tk jgh gSa ysfdu

Jksrk D;ksafd lzksr }kjk mRiUu igyh rjax ds ckn Lo;a vo T nwjh py pqdk

gksrk gSA vr% lzksr }kjk mRiUu nwljh rjax mls dqN le; igys vFkok ckn esa

lqukbZ iM+rh gSA mls ,slk vkHkkl gksrk gS ekuks rjaxsa cnys gq, osx ls mldh

vksj vk jgh gSaA

ekuk fd Jksrk lzksr dh vksj tk jgk gSA vr% lzksr }kjk mRiUu rjax dh rjaxnS/;Z% vT=λ lzksr }kjk mRiUu rjax dk osx% v Jksrk }kjk izkIr rjax dk osx% v+ vo

Jksrk }kjk izkIr rjax dh vko`fÙk% nv

vv

vT

vvvvn ooo )()(' +=+=+=

λ

,slh fLFkfr esa lzksr ,d gh LFkku ls rjaxksa dks mRiUu djrk gS ysfdu

Jksrk igyh rjax dks izkIr djus ds ckn og Lo;a vo T py pqdk gksrk gSA

ftlls Jksrk }kjk izkIr nks rjaxksa ds chp dk varjky cny tkrk gS vkSj og

vko`fÙk esa vkHkklh cnyko eglwl djrk gSA vxj Jksrk lzksr ls nwj tkrk gS rks

varjky c<+ tkrk gS ftlls vkHkklh vko`fÙk ?kV tkrh gSA ysfdu Jksrk ds lzksr

dh vksj vkus ij ;g varjky ?kV tkrk gS ftlls vkHkklh vko`fÙk c<+ tkrh gSA

nksuksa pyk;eku gks vc ge pyk;eku ek/;e ij fopkj djrs gSaA

113

bdkbZ&9 Å"ekfefr vkSj Å"ek lapj.kbdkbZ&9 Å"ekfefr vkSj Å"ek lapj.kbdkbZ&9 Å"ekfefr vkSj Å"ek lapj.kbdkbZ&9 Å"ekfefr vkSj Å"ek lapj.k

ikuh dk f=d fcUnq] M;wykax ,oa isfVV~ dk fu;e] izhoksLV dk Å"ek fofue; dk fu;e] rkih; izlkj] xSlksa dh fof'k"V Å"ek] LVhQu dk fu;e] U;wVu dk

'khryu dk fu;e

loZizFke f'k{kd Nk=ksa dks Å"ek] DoFkukad] xyukad Å"eh; izlkj] fof'k"V Å"ek

vkfn fcUnqvksa ij O;kogkfjd dbZ mnkgj.kksa dks izLrqr dj fo"k;oLrq dh vo/kkj.kk dks

Li"V djkosa rFkk bu fo"k; oLrqvksa dh vko';drk ,oa egRo dks crkrs gq, fo"k;oLrq

ij vk;sA

1111---- ikuh dk f=d fcUnq ikuh dk f=d fcUnq ikuh dk f=d fcUnq ikuh dk f=d fcUnq (Triple point of water) -

At one atmosphere pressure water is liquid between 0o to 100oC

inkFkZ dh fofHkUu voLFkk,a Qst dgykrh gSaA lkekU; rki vkSj nkc ij fofHkUu inkFkZ

fofHkUu Qst esa feyrs gSaA

,d ok;qe.Myh; nkc ij ty ds chp 0o ls 100oC ds chp nzo Qst ;kfu voLFkk esa

jgrk gSA ok;qe.Myh; nkc ij 100oC ij ty lkE;koLFkk esa nzo vkSj xSl Qst ;kfu

nks Qst esa jgrk gSA ;kfu bl rki ij fudk; dk ,d Hkkx nzo Qst esa vkSj 'ks"k xSl

Qst esa ,d lkFk jgrs gSaA blhrjg ok;qe.Myh; nkc ij 0oC ij ty lkE;koLFkk esa

nzo vkSj Bksl Qst ;kfu nks Qst esa jgrk gSA ;kfu bl rki ij fudk; dk ,d Hkkx

nzo Qst esa vkSj 'ks"k Bksl Qst esa ,d lkFk jgrs gSaA

mi;qZDr mnkgj.k ls Li"V gksrk gS fd lkE;koLFkk esa dksbZ Hkh fudk; ,d ;k ,d ls

vf/kd Qst esa jg ldrk gSA

bls le>us ds fy;s ge p-v vkjs[k dh lgk;rk ys ldrs gSaA p-v vkjs[k ij ty ds

le&rkih; odzksa ds v/;;u ls gesa ty ds fVªiy ikbZaV dh tkudkjh feyrh gSA ty

dk fVªiy ikbZaV og rki vkSj nkc gS tgk¡ ty ds rhuksa Qst ,d lkFk feyrs gSaA p-v

vkjs[k ij bl rki ij feyus okyh lerkih; nzo&xSl odz lerkih; nzo&xSl odz ls

coincide djrh gSA ty ds fy;s fVªiy ikbZaV 0-01 oC rFkk nkc 0-006 ok;qe.My

gksrk gSA

114

2222---- Mqykax isfVV fu;e Mqykax isfVV fu;e Mqykax isfVV fu;e Mqykax isfVV fu;e (Dulong Petit's law) &&&&

;g fu;e lkekU; rki ij Bkslksa dh fof'k"V Å"ek dks ekius gsrq fdls x;s iz;ksxksa ls

fudys fu"d"kksZa ij vk/kkfjr gSA bl fu;e ds vuqlkj lkekU; rki ij Bkslksa dh

ijek.kfod fof'k"V Å"ek ¼ijek.kq Hkkj × fof'k"V Å"ek /kkfjrk = 5-96 dSyksjh@fMxzh&

ijek.kfod Hkkj½ fu;r gksrh gSA dqN vioknksa ¼tSls ghjk] cksjku] flyhdkWu½ dks NksM+

dj lkekU; rki ij vf/kdrj Bkslksa ds fy;s ;g fu;e dke djrk gSA

Mqykax isfVV fu;e dh O;k[;k cksYV~teSu ds Lokra«; dksfV ds fu;e ds vk/kkj

ij dh tk ldrh gSA bldks le>us ds fy;s ijek.kqvksa dh xfr dks le>uk gksrk gSA

pw¡fd dksbZ Hkh ijek.kq x- fn'kk] y- fn'kk vFkok z- fn'kk esa LFkkukarj.kh; xfr djus esa

leFkZ gksrk gS vr% bldh rhu Lokra«; dksfV;k¡ gksrh gSaA cksYV~teSu ds Lokra«; dksfV

ds fu;e ds vuqlkj Lokra«; dksfV ds lkFk kT21 ÅtkZ lac) jgrh gSA blrjg rhu

Lokra«; dksfV;ksa ls lac) fdlh ijek.kq dh ÅtkZ kT23 gksxhA pw¡fd ijek.kq Hkkj esa

ijek.kqvksa dh la[;k ,oksxsMªks la[;k ¼n½ ds cjkcj gkrh gS vr% dqy ÅtkZ

nkTE23=

ijek.kfod fof'k"V Å"ek RnkdT

dE

23

23 ===

ckn esa fd;s x;s iz;ksxksa ls Kkr gqvk fd mPp vkSj fuEu rki ij ;g fu;e

dke ugha djrk gS D;ksafd bu rkiksa ij Bkslksa esa mifLFkr bysDVªkuksa dh xfr izHkkoh gks

tkrh gS vkSj bids ;ksxnku dks Hkh x.kuk esa lfEefyr djuk gksrk gSA

3333---- izksoksLV dk Å"ek fofue; fu;e izksoksLV dk Å"ek fofue; fu;e izksoksLV dk Å"ek fofue; fu;e izksoksLV dk Å"ek fofue; fu;e (Prevost's Theory of Exchange)

vko';d funsZ'kvko';d funsZ'kvko';d funsZ'kvko';d funsZ'k

izhoksLV ds fofue; fl)kar ds vuqlkj izR;sd oLrq lHkh rkiksa ij ¼ije 'kwU; rki

dks NksM+dj½ Å"eh; fofdj.k mRlftZr djrh gS rFkk og vius leheorhZ vU; oLrqvksa

Å"ek ds vknu&iznku ij vk/kkfjr O;kogkfjd mnkgj.k izÅ"ek ds vknu&iznku ij vk/kkfjr O;kogkfjd mnkgj.k izÅ"ek ds vknu&iznku ij vk/kkfjr O;kogkfjd mnkgj.k izÅ"ek ds vknu&iznku ij vk/kkfjr O;kogkfjd mnkgj.k izLrqr djsaLrqr djsaLrqr djsaLrqr djsa

tSls& B.M yxuk] xeZ oLrq dk B.Mk gksuk vkfnAtSls& B.M yxuk] xeZ oLrq dk B.Mk gksuk vkfnAtSls& B.M yxuk] xeZ oLrq dk B.Mk gksuk vkfnAtSls& B.M yxuk] xeZ oLrq dk B.Mk gksuk vkfnA

115

¼okrkoj.k½ ls mRlftZr fofdj.k dks vo'kksf"kr Hkh djrh gSA bl izdkj oLrq rFkk ckg~;

okrkoj.k ds chp Å"ek dk vknku&iznku rc rd tkjh jgrk gS tc rd dh mlesa

Å"eh; lkE; LFkkfir ugha gks tkrk gSA

;fn mRlftZr ÅtkZ dk eku vo'kksf"kr ÅtkZ ls vf/kd gksrk gS rks oLrq dk

rki ?kVrk gS vkSj ;fn mRlftZr ÅtkZ vo'kksf"kr ÅtkZ ls de gksrh gS rks oLrq dk rki

c<+rk gSA

4444---- rkih; izlkj rkih; izlkj rkih; izlkj rkih; izlkj (Thermal expansion) -

;g gekjs nSfud thou ds vusd vuqHkoksa ls laca/k j[krk gSA

tc fdlh Bksl dks xje fd;k tkrk gS rks mldk foLrkj lHkh vksj vR;ar

lefer rjhds ls gksrk gSA bl rjg Bkslksa esa rkih; izlkj ,sls gksrk gS tSls fdlh QksVks

dk bUyktZesaV gksrk gSA

bls le>us ds fy;s ge L yEckbZ dh ,d iryh NM+ ysrs gSaA iryh NM+ ds

p;u ls ge pkSM+kbZ vkSj eksVkbZ ij rkih; izlkj dks ux.; eku dj NksM+ ldrs

gSaA ,slk djus ls xf.krh; v/;;u lgt gks tkrk gSA fQj rkih; izlkj dh lefer

izd`fr dk lgkjk ys dj ge fdlh Hkh Bksl esa pkSM+kbZ vkSj eksVkbZ ij gksus okys rkih;

izlkj dks le> ldrs gSaA

vc xf.krh; n`f"V ls vkxs c<+rs gSaA

iryh NM+ esa gksus okys rkih; izlkj dks ge ^js[kh; rkih; izlkj ¼∆ L½* dgrs

gSaA vuqHko crkrk gS fd ;g izlkj inkFkZ dh izd`fr ds lkFk nks fuEu HkkSfrd izkpyksa ij

fuHkZj djrk gSA

1- yEckbZ L

2- rki esa of) ¼∆ T½

vr% ^js[kh; rkih; izlkj ¼∆ L½*dks mi;qZDr nksuksa HkkSfrd izkpyksa ds lekuqikfrd ekuus

ij TLL ∆∝∆

( )TLL ∆=∆ α

116

α dks ^js[kh; rkih; izlkj xq.kkad* dgrs gSa ftldk laca/k inkFkZ dh izd`fr ls

gksrk gSA α dk eku Hkh rki ds lkFk dqN u dqN cnyrk gS ysfdu iz;ksx crkrs gSa fd vf/kdrj O;ogkfjd fLFkfr;ksa esa bls fu;r ekuk tk ldrk gSA

lefefr ds vuqlkj&lefefr ds vuqlkj&lefefr ds vuqlkj&lefefr ds vuqlkj&

yEckbZ esa rkih; izlkj xq.kkad = pkSM+kbZZ esa rkih; izlkj xq.kkad = Å¡pkbZ esa rkih;

izlkj xq.kkad

nzoksa esa vk;ru izlkj gh egRoiw.kZ gksrk gSA ftl rjg geus ^js[kh; rkih; izlkj* ds

fy;s lw= izkIr fd;k Fkk] mlh rjg rkfdZd rjhds ls ge vk;ru izlkj ds fy;s Hkh lw=

izkIr dj ldrs gSaA

( )TVV ∆=∆ γ

γ dks ^vk;ru rkih; izlkj xq.kkad* dgrs gSa ftldk laca/k nzo dh izd`fr ls gksrk gSA nzoksa esa vk;ru izlkj ds v/;;u ds fy;s gesa mls fdlh cjru esa j[kuk gksrk

gSA xje djrs le; u flQZ nzo gh xje gksrk gS] cjru Hkh xje gksrk gS ftlls nksuksa

dk gh izlkj gksrk gSA vr% iz;ksfxd v/;;u ds nkSjku tks ekik tkrk gS og nzo dk

okLrfod izlkj ugha gksrk gSA tks ekik tkrk gS og vkHkklh izlkj gksrk gS ftls

la'kksf/kr dj nzo esa gq, okLrfod izlkj dks Kkr fd;k tkrk gSA

lefefr ds vk/kkj ij ^vk;ru rkih; izlkj xq.kkad* γ] ^js[kh; rkih; izlkj

xq.kkad* α dk frxquk gksrk gSA

vk;ru rkih; izlkj xq.kkad = yEckbZ esa rkih; izlkj xq.kkad + pkSM+kbZZ esa rkih;

izlkj xq.kkad + Å¡pkbZ esa rkih; izlkj xq.kkad

γ = 3α

5555---- xSlksa dh fof'k"V Å"ek& xSlksa dh fof'k"V Å"ek& xSlksa dh fof'k"V Å"ek& xSlksa dh fof'k"V Å"ek& Specific heat of gases

tc Hkh fdlh xSl dks Å"ek nh tkrh gS mlds nkc vkSj vk;ru esa o`f) gksrh

gSA vr% xSlksa dh fof'k"V Å"ek dk ekiu nks izdkj ls fd;k tk ldrk gSA ,d esa xSl

ds vk;ru dks fLFkj j[krs gq, Å"ek nsrs gq, rki esa o`f) ekih tkrh gS rFkk nwljs esa

xSl ds nkc dks fLFkj j[krs gq, Å"ek nsrs gq, rki esa o`f) ekih tkrh gSA igys izdkj

117

ls feyus okyh fof'k"V Å"ek dks ¼cv½ vkSj nwljs izdkj ls feyus okyh fof'k"V Å"ek dks

¼cp½ dgrs gSaA xSl ds ,d eksy ds rki esa ,d fMxzh dh o`f) esa yxus okyh Å"ek dh

ek=k dks fof'k"V Å"ek dgrs gSaA

tc xSl dk vk;ru fu;r jgrk gS rc xSl dks nh tk jgh lkjh Å"ek mlds

rki dks c<+kus esaa gh [kpZ gksrh gSA ysfdu tc xSl dk nkc fLFkj jgrk gS rc vk;ru

esa o`f) ¼∆V½ gksrh gS rc xSl dks vfrfjDr dk;Z Hkh djuk gksrk gSA vr% nh tk jgh

Å"ek xSl esa rki o`f) ds lkFk gh vfrfjDr dk;Z esa Hkh [kpZ gksrh gSA vr% Li"V gh

¼cp½] ¼cv½ ls cM+h gksuk pkfg;sA vkSj bu nksuksa esa varj dks ¼cp½ ekiu ds nkSjku fd;s

x;s vfrfjDr dk;Z esa [kpZ gqbZ Å"ek ds cjkcj gksuk pkfg;sA

¼cp½ - ¼cv½ = vfrfjDr dk;Z esa [kpZ gqbZ Å"ek

vfrfjDr dk;Z ¼vfrfjDr dk;Z ¼vfrfjDr dk;Z ¼vfrfjDr dk;Z ¼∆∆∆∆W½ dh x.kuk½ dh x.kuk½ dh x.kuk½ dh x.kuk

lkekU; lw=&

dk;Z = ¼cy½ ⋅ ¼foLFkkiu½

bl lw= dks xSlksa ds v/;;u gsrq nkc vkSj vk;ru esa cnyuk gksrk gSA

dk;Z = [¼cy@{ks=Qy½] ⋅ ¼{ks=Qy½⋅¼foLFkkiu½

[¼cy@{ks=Qy½] = nkc

¼{ks=Qy½⋅¼foLFkkiu½ = vk;ru

vr% fuf'pr nkc ij xSl }kjk fd;k x;k dk;Z = nkc ⋅ vk;ru esa ifjorZu

∆W = p ∆V

vkn'kZ xSl ds fy;s

p V = RT

p ∆V= R∆T

vkn'kZ xSl ds }kjk izfr fMxzh rki o`f) ¼∆T = 1½ esa fd;k x;k dk;Z R ds

cjkcj gksxkA Å"ekxfrdh ds igys fu;e ls JHW = gksrk gSA vr% JHR = gksxkA ;kfu]

vfrfjDr dk;Z djus esa vkn'kZ xSl }kjk [kpZ gqbZ Å"ek J

RH = gksxhA vr%

118

¼cp½ - ¼cv½ = J

R

6666---- LVhQu dk fu;e LVhQu dk fu;e LVhQu dk fu;e LVhQu dk fu;e (Stefan's Law) - vko';d funsZ'kvko';d funsZ'kvko';d funsZ'kvko';d funsZ'k

LVhQu ds fu;ekuqlkj fdlh iw.kZ d`".k fi.M ds ,dkad {ks=Qy ls izfr lsds.M

mRlftZr fofdj.k ÅtkZ] fi.M ds ijerki dh prqFkZ ?kkr ds vuqØekuqikrh gksrh gSA

;fn fdlh d`".k fi.M dk ijerki T gS rks mlds ,dkad i`"B ds {ks=Qy ls

izfr lssd.M mRlftZr ÅtkZ

E � T4 ;k E = 9 T 4

;gka ,d 9 fu;rkad gS ftls LVhQu dk fu;rkad dgrs gSA

S.I. i)fr esa bldk ek=d twy lsd.M&1 ehVj&2 K -4

mijksDr lw= ds vk/kkj ij vkafdd iz'u djk,¡ ,oa LVhQu cksYV~teSu fu;e ls

Hkh voxr djok,¡A

7777---- U;wVu dk 'khyru fu;e U;wVu dk 'khyru fu;e U;wVu dk 'khyru fu;e U;wVu dk 'khyru fu;e (Newton's Law of Cooling) -

vko';d funsZ'kvko';d funsZ'kvko';d funsZ'kvko';d funsZ'k

U;wVu ds 'khyu fu;ekuqlkj leku voLFkk jgus ij fofdj.k }kjk fdlh oLrq ds

B.Ms gksus dh nj ml oLrq vkSj vklikl ds okrkoj.k ds rkikarj ds vuqØekuqikrh

gksrh gSA ¼tcfd rkikarj cgqr vf/kd u gks½

oLrq ls Å"ek gkfu dh nj vFkok oLrq ds B.Ms gksus dh nj � rkikarj ;fn fdlh

oLrq dk izkjafHkd rki �10C gS rFkk t lsd.M ckn mldk rki �2

0C gks tkrk gS]

rks oLrq ds B.Ms gksus dh nj ¾ �10C - �2

0C

t

d".k fi.M] ijerki] ÅtkZ fofdj.k] ÅtkZ gkfu dh d".k fi.M] ijerki] ÅtkZ fofdj.k] ÅtkZ gkfu dh d".k fi.M] ijerki] ÅtkZ fofdj.k] ÅtkZ gkfu dh d".k fi.M] ijerki] ÅtkZ fofdj.k] ÅtkZ gkfu dh

vo/kkj.kkvksa dks igys Nk=ksa dks Li"V djok,A vo/kkj.kkvksa dks igys Nk=ksa dks Li"V djok,A vo/kkj.kkvksa dks igys Nk=ksa dks Li"V djok,A vo/kkj.kkvksa dks igys Nk=ksa dks Li"V djok,A

nSfud thou ls lacaf/kr mnkgj.k tSls& mPprki okyh oLrqvksa dk tYnh B.Mk gksuk] xeZ nSfud thou ls lacaf/kr mnkgj.k tSls& mPprki okyh oLrqvksa dk tYnh B.Mk gksuk] xeZ nSfud thou ls lacaf/kr mnkgj.k tSls& mPprki okyh oLrqvksa dk tYnh B.Mk gksuk] xeZ nSfud thou ls lacaf/kr mnkgj.k tSls& mPprki okyh oLrqvksa dk tYnh B.Mk gksuk] xeZ

ty dk ty dk ty dk ty dk B.M esa tYnh B.Mk gksuk xehZ esa nsj ls B.Mk gksuk vkfn mnkgj.kksa ls fu;eksa dh B.M esa tYnh B.Mk gksuk xehZ esa nsj ls B.Mk gksuk vkfn mnkgj.kksa ls fu;eksa dh B.M esa tYnh B.Mk gksuk xehZ esa nsj ls B.Mk gksuk vkfn mnkgj.kksa ls fu;eksa dh B.M esa tYnh B.Mk gksuk xehZ esa nsj ls B.Mk gksuk vkfn mnkgj.kksa ls fu;eksa dh

vo/kkj.kk dks Li"V djsaA vo/kkj.kk dks Li"V djsaA vo/kkj.kk dks Li"V djsaA vo/kkj.kk dks Li"V djsaA

119

rFkk oLrq dk ek/;rki ¾ �10C + �2

0C

2

vc ;fn oLrq ds pkjksa vkSj okrkoj.k dk rki � 0C gS rks oLrq ds ek/;rki rFkk

okrkoj.k ds rki esa vUrj ¾ �10C + �2

0C - � 0C

2

vr% U;wVu ds 'khryu fu;e ls �10C - �2

0C � �10C + �2

0C - � 0C

t 2

'khryu dh lhek,¡&'khryu dh lhek,¡&'khryu dh lhek,¡&'khryu dh lhek,¡&

1- oLrq ds rki rFkk mlds pkjksa vkSj ds okrkoj.k ds rki esa vUrj vf/kd ugha

gksuk pkfg,A

2- oLrq }kjk Å"ek dh gkfu dsoy fofdj.k }kjk gh gksuh pkfg,A

3- oLrq dh izd`fr rFkk mldk {ks=Qy ugha cnyuk pkfg,A

129

xSl lehdj.kxSl lehdj.kxSl lehdj.kxSl lehdj.k vkn'kZ xSl lehdj.k

RTpv = okLrfod xSl lehdj.k

( ) RTbvv

ap =−

+ 2

vkn'kZ xSl vkSj okLrfod xSl esa varj dks tkus cxSj bu lehdj.kksa dks le>kus esa dfBukbZ vkrh gSA vkn'kZ xSl esa tgk¡ xSl ds v.kqvksa ds vk;ru vkSj vkSj mlds v.kqvksa ds chp dk;Zjr varvkZ.kfod cy dks ux.; ekurs gq, NksM+ fn;k tkrk gS ogha okLrfod xSl esa bu ij fopkj fd;k tkrk gSA

vkn'kZ xSl esa xSl ds v.kqvksa ds vk;ru dks ux.; ekuus ds dkj.k xSl dk vk;ru ml crZu ds vk;ru ds cjkcj ekuk tkrk gS ftlesa mls j[kk tkrk gSA ysfdu okLrfod xSl esa ,slk ekuuk mfpr ugha gksrk gSA pw¡fd okLrfod xSl esa v.kqvksa ds vk;ru dks ux.; ugha ekuk tk ldrk vr% xSl dk okLrfod vk;ru ml crZu ds vk;ru ls dqN de gksuk pkfg;s ftlesa mls j[kk tkrk gSA bls vk;ru la'kks/ku vk;ru la'kks/ku vk;ru la'kks/ku vk;ru la'kks/ku dgrs gSaA

vc ge xSl ds v.kqvksa ds chp dk;Zjr varvkZ.kfod cyksa ij fopkj djrs gSA vxj ;g varvkZ.kfod cy ugha gksrk rks gesa fuEu rki ij vkSj mfpr ifjfLFkfr;ksa esa dksbZ Hkh xSl nzo curs gq, ugha feyrhA pw¡fd fdlh Hkh okLrfod xSl dk nzohdj.k laHko gksrk gS vr% xSl ds v/;;u ds nkSjku varvkZ.kfod cyksa dks ux.; ugha ekuk tk ldrkA okLrfod xSl esa varvkZ.kfod cyksa dh mifLFkfr ds dkj.k vkn'kZ xSl dh rqyuk esa nkc dk eku dqN vf/kd gks tkrk gSA bls nkc la'kks/ku nkc la'kks/ku nkc la'kks/ku nkc la'kks/ku dgrs gSaA

ok.Mjoky us okLrfod xSlksa ds v/;;u ds nkSjku bUgha nksuksa la'kks/kuksa ij fopkj djrs gq, vkn'kZ xSl lehdj.k dks la'kksf/kr fd;kA

ok.Mjoky us vk;ru la'kks/ku vk;ru la'kks/ku vk;ru la'kks/ku vk;ru la'kks/ku ij fopkj djrs gq, xSl ds okLrfod vk;ru crZu ds vk;ru ls ^b* de ekukA muds vuqlkj ^b* ,d ,slk fu;rkad gS ftldk eku xSl ds xq.kksa ij fuHkZj djrk gSA vc ok.Mjoky us nkc la'kks/kunkc la'kks/kunkc la'kks/kunkc la'kks/ku ij fopkj fd;kA nkc cjru dh nhokjksa ij xSl ds v.kqvksa ds Vdjkus ls mRiUu gksus okys cy ds dkj.k feyrk gSA cjru dh nhokj ds izfr bZdkbZ {ks=Qy ij yxus okys cy dks nkc dgrs gSaA vr% ;g nkc fdlh Hkh le; cjru dh nhokjksa ds ij bZdkbZ {ks=Qy ij mifLFkr v.kqvksa ds lekuqikfrd rks gksuk gh pkfg;sA

130

ysfdu cjru esa mifLFkr 'ks"k v.kq nhokjksa ij mifLFkr v.kqvksa dks varvkZ.kfod cyksa ds dkj.k vanj gh vksj vkdf"kZr djsaxs ftlls nkc esa o`f) gksxhA bldk eryc ;g gqvk fd nkc esa o`f) cjru dh nhokjksa ds ij bZdkbZ {ks=Qy ij mifLFkr v.kqvksa vkSj cjru esa mifLFkr 'ks"k v.kqvksa dh la[;k ij fuHkZj djsxhA ;s nksuksa gh la[;k,a cjru esa xSl ds vk.kfod ?kuRo ij fuHkZj djrh gSaA vr% varvkZ.kfod cyksa ds dkj.k nkc esa o`f) vk.kfod ?kuRo ds oxZ ds lekuqikfrd gksxhA pw¡fd vk.kfod ?kuRo cjru ds vk;ru dk O;qRdzekuqikfrd gksrk gS] vr% ok.Mjoky us nkc la'kks/ku dks ^a/v2* ds :i esa vfHkO;Dr fd;kA ^b* dh gh rjg ^a* Hkh ,d fu;rkad gS ftldk eku xSl ds xq.kksa ij fuHkZj djrk gSA Å"ekxfrdh; izfØ;k,¡ Å"ekxfrdh; izfØ;k,¡ Å"ekxfrdh; izfØ;k,¡ Å"ekxfrdh; izfØ;k,¡ (Thermodynamic processes) -

Å"ekxfrdh; izfØ;kvksa ds v/;;u esa fudk; vkSj ftl okrkoj.k esa mls j[kk tkrk gS] mudh le> cgqr t:jh gksrh gSA fudk; vkSj okrkoj.k ds rki rFkk muds chp ds mifLFkr Hksnd dh izd`fr ls r; gksrk gS fd izfØ;k ds nkSjku fudk; vkSj okrkoj.k ds chp Å"ek vFkok nzO; dk vknku iznku gksrk gS ;k fudk; ds rki] vk;ru vkSj nkc esa ifjorZu gksrk gSA blh ls ;g Hkh r; gksrk gS fd fudk; ij dk;Z fd;k tk jgk gS vFkok fudk; dk;Z dj jgk gSA fudk; fudk; fudk; fudk;

fudk; rhu rjhds ds gksrs gSa% 1- foyx fudk;% fudk; vkSj okrkoj.k ds chp Å"ek vFkok nzO; fdlh dk Hkh

vknku iznku ugha gksrk gSA

2- can fudk;% fudk; vkSj okrkoj.k ds chp flQZ Å"ek dk vknku iznku gks

ldrk gSA

3- [kqyk fudk;% fudk; vkSj okrkoj.k ds chp Å"ek vFkok nzO; nksuksa dk gh

vknku iznku gks ldrk gSA

Hksnd dhHksnd dhHksnd dhHksnd dh izd`fr% izd`fr% izd`fr% izd`fr%

Hksnd dh izd`fr dqpkyd ¼Å"ek ds vknku iznku esa ck/kd½ vFkok lqpkyd ¼Å"ek ds vknku iznku esa lgk;d½ izdkj dh gks ldrh gSA Hksnd tkyhuqek ¼Å"ek vFkok nzO; ds vknku iznku esa lgk;d½ Hkh gks ldrk gSA

131

fudk; vkSj dk;Z%fudk; vkSj dk;Z%fudk; vkSj dk;Z%fudk; vkSj dk;Z%

fudk; }kjk fd;k x;k dk;Z /kukRed vkSj fudk; ij fd;k x;k dk;Z

_.kkRed gksrk gSA fudk; vkSj Å"ek%fudk; vkSj Å"ek%fudk; vkSj Å"ek%fudk; vkSj Å"ek%

fudk; dks Å"ek nsus ls nks ckrsa gks ldrh gSaA ;k rks fudk; ds rki esa

o`f) gks ldrh gS ;k fQj fudk; dqN cká dk;Z Hkh dj ldrk gSA ;k fQj fudk; ds lkFk nksuksa gh ckrsa gks ldrh gSaA fudk; ls Å"ek fudkyus ij fudk; ds rki esa deh gksrh gSA

tc fudk; vkSj okrkoj.k ds chp fdlh izdkj ls Å"ek dk vknku iznku ugha gksrk gS rks fQj nks ckrsa gks ldrh gSaA vxj fudk; ij dqN cká dk;Z fd;k tkrk gS rks mlds rki esa o`f) gksrh gSA vkSj vxj fudk; dqN cká dk;Z djrk gS rks mlds rki esa deh gksrh gSA

Å"ekxfrdh; izfØ;kvksa ds v/;;u esa Å"ekxfrdh ds izFke fu;e dk vuqiz;ksx fd;k tkrk gSA ;g fu;e Å"ek ifjorZu] fudk; dh vkarfjd ÅtkZ ¼rki ls lacaf/kr½ esa cnyko vkSj lEiUu dk;Z esa laca/k dks n'kkZrk gSA

dWdUdQ += pdVdUdQ +=

Å"ekxfrdh; izfØ;kvksa dk v/;;u fofHkUu ifjfLFkfr;ksa esa fd;k tk ldrk gSA mnkgj.k ds fy;s]

1- levk;rfud izfØ;k% 0=dV 2- lenkch; izfØ;k% 0=dp 3- lerkih; izfØ;k% 0=dT 4- :)ks"e izfØ;k% 0=dQ Å"ekxfrdh; izfØ;kvksa dks vkSj Hkh vU; rjhdksa ls ck¡Vk tk ldrk gSA ;Fkk] 1- mRdze.kh; izfØ;k 2- vuqRØe.kh; izfØ;k blhrjg] 1- pØh; izfØ;k

132

2- vpØh; izfØ;k Triple point of water At one atmosphere pressure water is liquid between 0o to 100oC

inkFkZ dh fofHkUu voLFkk,a Qst dgykrh gSaA lkekU; rki vkSj nkc ij fofHkUu inkFkZ fofHkUu Qst esa feyrs gSaA

,d ok;qe.Myh; nkc ij ty ds chp 0o ls 100oC ds chp nzo Qst ;kfu voLFkk esa jgrk gSA ok;qe.Myh; nkc ij 100oC ij ty lkE;koLFkk esa nzo vkSj xSl Qst ;kfu nks Qst esa jgrk gSA ;kfu bl rki ij fudk; dk ,d Hkkx nzo Qst esa vkSj 'ks"k xSl Qst esa ,d lkFk jgrs gSaA blhrjg ok;qe.Myh; nkc ij 0oC ij ty lkE;koLFkk esa nzo vkSj Bksl Qst ;kfu nks Qst esa jgrk gSA ;kfu bl rki ij fudk; dk ,d Hkkx nzo Qst esa vkSj 'ks"k Bksl Qst esa ,d lkFk jgrs gSaA

mi;qZDr mnkgj.k ls Li"V gksrk gS fd lkE;koLFkk esa dksbZ Hkh fudk; ,d ;k ,d ls vf/kd Qst esa jg ldrk gSA

bls le>us ds fy;s ge p-v vkjs[k dh lgk;rk ys ldrs gSaA p-v vkjs[k ij ty ds le&rkih; odzksa ds v/;;u ls gesa ty ds fVªiy ikbZaV dh tkudkjh feyrh gSA ty dk fVªiy ikbZaV og rki vkSj nkc gS tgk¡ ty ds rhuksa Qst ,d lkFk feyrs gSaA p-v vkjs[k ij bl rki ij feyus okyh lerkih; nzo&xSl odz lerkih; nzo&xSl odz ls coincide djrh gSA ty ds fy;s fVªiy ikbZaV 0-01 oC rFkk nkc 0-006 ok;qe.My gksrk gSA

133

Flow charts

tc Hkh dksbZ oLrq dEiu djrh gS rks ge /ofu lqurs gSaA bldk eryc gS fd /ofu dk laca/k dEiu ls gSA oLrq ftl ek/;e ¼gok] ikuh vkfn½ esa dEiu djrh gS mlesa fo{kksHk iSnk gksus yxrk gS ftlds dkj.k mlds d.k Hkh dEiu djus yxrs gSaA blls ek/;e esa rjax curh gS vkSj /ofu lzksr ls vkxs c<+us yxrh gSA tc ;g rjax gekjs dku ls Vdjkrh gS rks dku ds ijns dEiu djus yxrs gSa vkSj gesa /ofu lqukbZ nsus yxrh gSA

oLrq fdl rjg dEiu djrh gS\ dEiu djrs le; mldh xfr fdl fu;e ls lapkfyr gksrh gS\ dEiu vkSj rjax dh dkSulh pfj=xr fo'ks"krk,¡ lkeus vkrh gSa\ dEiu vkSj rjax dks le>us esa fdl rjg xf.kr dh lgk;rk yh tk ldrh gSA fdlh Hkh ,d fcanq ls lkspuk vkjaHk dj nhft;s vkSj vkxs c<+rs pys tkb;sA vkidh ftKklk ft/kj ys tk;s m/kj pyrs pys tkb;sA vki ns[ksaxs fd vius vki ,d dkalsIV esi ;kfu ¶yks pkVZ curk pyk tk;sxkA ftruh ckj lkspsaxs mruh gh ckj vyx&vyx izdkj ds ¶yks pkVZ cusaxsA gj ,d vius vki esa [kkl vkSj u;kA ;g fugk;r O;fDrxr vkSj vuqHkoksa ij vk/kkfjr gksrk gSA c<+rs vuqHko ds lkFk ;g csgrj curk pyk tkrk gSA uewus ds fy;s ;gk¡ dEiu vkSj rjax ls lacaf/kr ,d ¶yks pkVZ fn;k tk jgk gSA

Bodies

vibrates

Vibrations

Longitudinal Transverse

Free Forced

Undamped Damped

Driven

Resonance

Creates disturbance in media

Disturbance propogates as a wave

Progressive wave

Equation

Superposition

Stationary waves Interference Beats

Stretched string Air columns

Simple harmonic

Equation of motion

VelocityAcceleration

EnergyForce

Spring

Pendulum SimpleCompoundTorsional

Coupling

Coupled oscillator

Normal modes

Charged Neutral

MechanicalElectromagnetic

Longitudinal Transverse

Anharmonic

Displacement

Use calculus

Natural frequency

¡¢¶yks pkVZ Øekad &1

134

pyks vc ge inkFkZ vkSj fi.Mksa dh ckr djrs gSaA fi.M inkFkZ ls gh cus gksrs gSaA inkFkZ Bksl vkSj rjy nksuksa gh :i esa feyrs gSaA buesa tM+Ro] izR;kLFkrk] ';kurk] i`"Bruko vkfn lkekU; xq.kksa ds vfrfjDr Å"eh;] izdk'kh;] pqEcdh; vkSj fo|qrh; tSls xq.k Hkh feyrs gSaA bu xq.kksa dk vk/kkj inkFkZ dh vkarfjd lajpuk esa fNik gksrk gSA inkFkZ dh vkarfjd lajpuk esa v.kq vkSj ijek.kq rFkk muds chp lfdz; cy gksrs gSaA fdlh Hkh d.k ;k fi.M dks fudk; ekuk tkrk gSA fdlh Hkh LFkku vkSj le; ij fudk; dh fLFkfr dks ifjHkkf"kr djus ds fy;s funsZ'k ra= dk lgkjk ysuk iM+rk gSA fudk; ;k funsZ'k ra=] nksuksa esa ls dksbZ ,d vFkok nksuksa gh fojke ;k xfreku] fdlh Hkh voLFkk esa gks ldrs gSA xfr ;k fojke dh voLFkk lkis{kh; gksrh gSaA blrjg xfr] fudk; ;k funsZ'k ra= dksbZ Hkh gekjs ¶yks pkVZ dk vkjafHkd fcanq gks ldrk gSA vc lksprs&lksprs ge xfr;ksa ds izdkj] funsZ'k ra= ds Lo:i vkfn fo"k;ksa ij ftKklq gksrs gq, c<+us yxrs gSaA xfr ls tqM+h jkf'k;ksa tSls foLFkkiu] osx] cy vkfn fnekx esa vkus yxrs gSa vkSj /khjs /khjs ,d ¶yks pkVZ cuus yxrk gSA ftl fn'kk esa vkidks vf/kd :fp gS] ml fn'kk esa /;ku yxkb;sA jkLrk Lor% cuus yxrk gSA uewus ds fy;s ;gk¡ inkFkZ vkSj xfr ls lacaf/kr ,d ¶yks pkVZ fn;k tk jgk gSA

Electron Nucleus ProtonNeutron

QuarksAtoms

Matter

Intermolecular forces

Inter-atomic forces MoleculesNatural

Artificial

92

26

Bodies

Properties

GeneralThermalOpticalElectricalMmagnetic

System One-D, 2-D, 3-D

Frame of reference

InertialNon-inertial

Rest Motion

Constant velocity

Accelerated

Translatory Rotational Oscillatory

State

Motion

Newton'sLaws of motion

ForceAccelerationVelocityDisplacement

ClassificationNormalVelocities

Very largeVelocities

Equationsof motion

Newton'sLaws of motion fails

Einstein's theory

Translatory Rotational Oscillatory

Use of Calculusand vector analysis

Description requires

CollisionSolids

Fluids

ThermalProperties

Termodynamics

Elastic Inelastic

Conservation laws

E = mc2

¡¢¶yks pkVZ Øekad &2

135

blhrjg xq:Rokd"kZ.k ds fy;s Hkh ge ,d ¶yks pkVZ cuk ldrs gSaA bldk vkjaHk

mu ?kVukvksa ls gks ldrk gS ftuds fy;s ,d gh fu;e dke djrk gSA vkSj og

gS U;wWVu dk xq:Rokd"kZ.k dk fu;eA vc tc lksprs&lksprs vki ftKklq gksrs

gq, c<+us yxrs gSa rks dbZ phtsa vkids fnekx esa vkus yxrh gSa vkSj /khjs /khjs ,d

¶yks pkVZ cuus yxrk gSA ftl fn'kk esa vkidks vf/kd :fp gS] ml fn'kk esa

/;ku yxkb;sA jkLrk Lor% cuus yxrk gSA uewus ds fy;s ;gk¡ xq:Rokd"kZ.k ds

izHkko dks le>us gsrq ,d ¶yks pkVZ fn;k tk jgk gSA

Falling of fruits /leavesProjectile's motion

Motion of moon around earth

Planetary motion Tides

Governed by single law

Newton's law of gravitation

Sattelites GeostationaryPolarRemote sensingWeather

Force between two mass particles separated by a distance 'd'

FmM = G m M

d2= - FMm

m g

g =G M

d2

Accelearation due to gravity

gpoles > gequator

Kepler's laws

One- D motion

Two-D motion

Resolve the motion

Horizontal

Vertical

Vertical

Earth

Sattelite's path /Orbit

Equations of motion

Moon

Pulls ocean's water and creates tides

Ocean

Natural

Artificial

t:jh ugha fd ¶yks pkVZ cM+k gh gksA ;g vko';drkuqlkj NksVk ;k cM+k fdlh Hkh izdkj dk gks ldrk gSA ¶yks pkVZ dk mn~ns'; lksp esa foLrkj ykuk vkSj ;kn j[kusa esa vklkuh ykuk gSA uewus ds fy;s ;gk¡ rjy dks le>us gsrq ,d ¶yks pkVZ fn;k tk jgk gSA

vc ge ,d vkSj ¶yks pkVZ dks fodflr djsaxsA

;g Å"ekxfrdh ls lacaf/kr gSA gekjs nSfud

thou ds vf/kdrj vuqHkoksa dk laca/k Å"ek ds

¡¢¶yks pkVZ Øekad &3

¡¢¶yks pkVZ Øekad &4

Flow

Viscous Nonviscous

Turbulent(Rotational)

Steady (Streamline)

Compressible Incompressible

Fluid

Liquids GasesSurface

Surface tension

136

vknku&iznku dks ys dj gksrk gSA Å"ek dk ;g vknku&iznku fofHkUu izdkj ls

gks ldrk gSA Å"ek ls dk;Z vkSj dk;Z ls Å"ek dh izkfIr ls ge ifjfpr gSa ghA

jxM+us ;kfu ?k"kZ.k ls Å"ek mRifRr laca/kh vuqHko yxHkx lcdks gSA blrjg

Å"ek dk xfrdh ls xgjk laca/k gSA buls lacaf/kr fudk; Å"ekxfrdh; fudk;

dgykrs gSaA bu fudk;ksa dks rki] nkc] vk;ru tSlh HkkSfrd jkf'k;ksa ls igpkuk

tkrk gSA ;s HkkSfrd jkf'k;k¡ gh Å"ekxfrdh; funsZ'kkad dgykrh gSaA vc vkidk

vkjafHkd fcanq dqN Hkh gks ldrk gSA ftl fdlh dks Hkh vki vkjafHkd fcanq cuk;sa]

ogha ls vki lkspuk 'kq: dj nhft;sA vxj vki Å"ekxfrdh; fudk; ls vkxs

c<+uk pkgrs gSa rks vki lkspsaxs fd dSls ifjHkkf"kr djsa bls\ ifjHkkf"kr djus ds

i'pkr bldh voLFkk vkSj voLFkk ifjorZu dks ysdj vkids eu esa LokHkkfod

ftKklk mBsxhA blrjg lksprs lksprs vki vkxs c<+ ldrs gSaA rki&o`f) ls

inkFkZ ds dbZ xq.kksa esa Hkkjh cnyko vkrk gSA ;g cnyko rkiekih dk vk/kkj

curk gSA ;g Hkh vkjafHkd fcanq gks ldrk gSA vki Lo;a ,d ¶yks pkVZ cukb;sA

uewus ds fy;s ;gk¡ Å"ekxfrdh dks le>us gsrq ,d ¶yks pkVZ fn;k tk jgk gSA

Daily life experiences

Output < Input

Heat flowW <====> H

Thermodynamics

System

DynamicsHeat

Quantity

Quality

Temperature

Thermometry

ThermalExpansionVolumePressureResistance

T dependent Parameter

Closed Isolated Open

Macro Micro

Description

Thermodynamic quantities

P, V, T etc.

State

pv =RT

(p+a/v2) (v-b) =RT

Indicator diagram Change /Processes

Constancy of some parameters

Condition

IsothermalIsobaricIsochoricIsothermalAdiabatic

Reversible/IrreversibleStable/UnstableCyclic/noncyclicStatic/dynamic/QuasistaticThermodymamic equilibrium

Understandingat molecular level

Equation of state

Hg Thermometer

Air Thermometer

Alcohol Thermometer

Pt-Resistance ThermometerThermocouple

Thermo emf

Energy

Q = msT

========> <==========

¡¢¶yks pkVZ Øekad &5