1 - measurement

MEASUREMENT

Measurement in everyday life

Measurement of mass Measurement of volume

Measurement in everyday life

Measurement of length Measurement of temperature

Need for measurement in physics

• To understand any phenomenon in physics we have to perform experiments.

• Experiments require measurements, and we measure several physical properties like length, mass, time, temperature, pressure etc.

• Experimental verification of laws & theories also needs measurement of physical properties.

Physical QuantityA physical property that can be measured and

described by a number is called physical quantity.Examples:• Mass of a person is 65 kg.• Length of a table is 3 m.• Area of a hall is 100 m2.• Temperature of a room is 300 K

Types of physical quantities1.Fundamental quantities:

The physical quantities which do not depend on any other physical quantities for their measurements are known as fundamental quantities.Examples:• Mass• Length

• Time• Temperature

Types of physical quantities

The physical quantities which depend on one or more fundamental quantities for their measurements are known as derived quantities.

Examples:• Area• Volume

• Speed• Force

2. Derived quantities:

Units for measurementThe standard used for the measurement of

a physical quantity is called a unit.Examples:• metre, foot, inch for length• kilogram, pound for mass• second, minute, hour for time• fahrenheit, kelvin for temperature

Characteristics of unitsWell – definedSuitable sizeReproducible

InvariableIndestructible

Internationally acceptable

• This system was first introduced in France.

• It is also known as Gaussian system of units.

• It is based on centimeter, gram and second as the fundamental units of length, mass and time.

CGS system of units

MKS system of units

• This system was also introduced in France.

• It is also known as French system of units.• It is based on meter, kilogram and

second as the fundamental units of length, mass and time.

FPS system of units

• This system was introduced in Britain.• It is also known as British system of units.• It is based on foot, pound and second as

the fundamental units of length, mass and time.

International System of units (SI)

• In 1971, General Conference on Weight and Measures held its meeting and decided a system of units for international usage.

• This system is called international system of units and abbreviated as SI from its French name.

• The SI unit consists of seven fundamental units and two supplementary units.

Seven fundamental unitsFUNDAMENTAL QUANTITY SI UNIT SYMBOL

Length metre mMass kilogram kgTime second s

Temperature kelvin KElectric current ampere A

Luminous intensity candela cdAmount of substance mole mol

Definition of metre

The metre is the length of the path travelled by light in a

vacuum during a time interval of1/29,97,92,458 of a second.

Definition of kilogram

The kilogram is the mass of prototype cylinder of platinum-iridium alloy

preserved at the International Bureau of Weights and Measures, at Sevres,

near Paris.

Prototype cylinder of platinum-iridium alloy

Definition of second

One second is the time taken by 9,19,26,31,770 oscillations of the

light emitted by a cesium–133 atom.

Two supplementary units1.Radian: It is used to measure plane

angle

= 1 radian

Two supplementary units2. Steradian: It is used to measure solid

angle

= 1 steradian

Rules for writing SI units

1Full name of unit always starts with small

letter even if named after a person.• newton• ampere• coulom

b

not• Newton• Ampere• Coulom

b

Rules for writing SI units

2Symbol for unit named after a scientist

should be in capital letter.

• N for newton• K for kelvin

• A for ampere• C for coulomb

Rules for writing SI units

3Symbols for all other units are written in

small letters.

• m for meter• s for second

• kg for kilogram• cd for candela

Rules for writing SI units

4One space is left between the last digit of

numeral and the symbol of a unit.• 10 kg• 5 N• 15 m

not• 10kg• 5N• 15m

Rules for writing SI units

5The units do not have plural forms.• 6 metre• 14 kg• 20

second• 18 kelvin

not• 6 metres• 14 kgs• 20 seconds• 18 kelvins

Rules for writing SI units

6Full stop should not be used after the

units.• 7 metre• 12 N• 25 kg

not• 7 metre.• 12 N.• 25 kg.

Rules for writing SI units

7No space is used between the symbols for

units.• 4 Js• 19 Nm• 25 VA

not• 4 J s• 19 N m.• 25 V A.

SI prefixesFactor Name Symbol Factor Name Symbol

yotta Y deci dzetta Z centi cexa E milli mpeta P micro μtera T nano ngiga G pico p

mega M femto fkilo k atto a

hecto h zepto zdeka da yocto y

• 3 milliampere = 3 mA = 3 x A• 5 microvolt = 5 μV = 5 x V• 8 nanosecond = 8 ns = 8 x s• 6 picometre = 6 pm = 6 x m• 5 kilometre = 5 km = 5 x m• 7 megawatt = 7 MW = 7 x W

Use of SI prefixes

Some practical units for measuring length1 micron = m

Bacterias Molecules

1 nanometer = m

Some practical units for measuring length1 angstrom = m

Atoms Nucleus

1 fermi = m

Some practical units for measuring length• Astronomical unit = It is defined as the mean distance

of the earth from the sun.• 1 astronomical unit = m

Distance of planets

Some practical units for measuring length• Light year = It is the distance travelled by light in vacuum

in one year.• 1 light year = m

Distance of stars

Some practical units for measuring length• Parsec = It is defined as the distance at which an arc of 1

AU subtends an angle of 1’’.• It is the largest practical unit of distance used in

astronomy.• 1 parsec = m

1 AU 1”

1 parsec

Some practical units for measuring area• Acre = It is used to measure large areas in British system

of units. 1 acre = 208’ 8.5” x 208’ 8.5” = 4046.8

• Hectare = It is used to measure large areas in French system of units.

1 hectare = 100 m x 100 m = 10000

• Barn = It is used to measure very small areas, such as nuclear cross sections.

1 barn =

Some practical units for measuring mass1 metric ton = 1000

kg

Steel bars Grains

1 quintal = 100 kg

1 pound = 0.454 kg

Newborn babies

Crops

1 slug = 14.59 kg

Some practical units for measuring mass

Some practical units for measuring mass• 1 Chandrasekhar limit = 1.4 x mass of sun = kg• It is the biggest practical unit for measuring mass.

Massive black holes

Some practical units for measuring mass

• 1 atomic mass unit = x mass of single C atom

• 1 atomic mass unit = 1.66 kg• It is the smallest practical unit for

measuring mass.• It is used to measure mass of single atoms,

proton and neutron.

Some practical units for measuring time

• 1 Solar day = 24 h

• 1 Sidereal day = 23 h & 56 min

• 1 Solar year = 365 solar day = 366 sidereal day

• 1 Lunar month = 27.3 Solar day

• 1 shake = s

Seven dimensions of the worldFundamental

quantitiesLength MassTime

TemperatureCurrent

Amount of substanceLuminous intensity

Dimensions[L] [M][T] [K][A][N][J]

Dimensions of a physical quantity

The powers of fundamental quantities in a derived quantity are called dimensions of that quantity.

¿Masslength×breath×   height

[Density ]= [M ][ L ]× [ L ]× [ L ]

=[M ][L3 ]

=[M L−3]

Dimensions of a physical quantity

Density=MassVolume

Example :

H ence the dimensions of density are1∈mass∧−3∈length .

Uses of Dimension

To check the correctness of equation

To convert units

To derive a formula

To check the correctness of equation

∆ x=displacement=[L]

Consider the equationof displacement ,

B y writing the dimensionswe get ,

v i t=velocity × time=lengthtime ×time=[L]

a t2=acceleration×t ime2=l engthtime 2

×time2=[L]

The dimensions of each term are same, hence the equation is dimensionally correct.

∆ x=v i t+12 a t

2

To convert unitsLet us convert newton (SI unit of force )into dyne (CGS unit of force ) .

T hedimesions of force are=[LM T −2]

So ,1newton=(1m)(1kg)(1 s)− 2

a nd ,1dyne=(1cm)(1g)(1 s)−2

T hus , 1newton1dyne

=( 1m1cm )( 1kg1g )( 1 s1 s )− 2

=( 100 c m1cm )( 1000g1g )( 1 s1 s )− 2

Therefore,

To derive a formulaThe time period ‘T’ of oscillation of a simple pendulum depends on length ‘l’ and acceleration due to gravity ‘g’.

Let us assume that,

T or T

K = constant which is dimensionless

Dimensions of T

Dimensions of

Dimensions of g

Thus,

[L0M 0T1 ]=K [La+bM 0T −2b ]

a+b=0∧−2b=1

∴b=− 12∧a=12

T

T

Least count of instruments

The smallest value that can be measured by the measuring instrument

is called its least count or resolution.

LC of length measuring instrumentsRuler scale

Least count = 1 mm

Vernier Calliper

Least count = 0.1 mm

LC of length measuring instrumentsScrew Gauge

Least count = 0.01 mm

Spherometer

Least count = 0.001 mm

LC of mass measuring instrumentsWeighing scale

Least count = 1 kg

Electronic balance

Least count = 1 g

LC of time measuring instrumentsWrist watch

Least count = 1 s

Stopwatch

Least count = 0.01 s

Accuracy of measurementIt refers to the closeness of a measurement

to the true value of the physical quantity.Example:• True value of mass = 25.67 kg• Mass measured by student A = 25.61 kg• Mass measured by student B = 25.65 kg• The measurement made by student B is more

accurate.

Precision of measurementIt refers to the limit to which a physical

quantity is measured.Example:• Time measured by student A = 3.6 s• Time measured by student B = 3.69 s• Time measured by student C = 3.695 s• The measurement made by student C is most

precise.

Significant figures

The total number of digits(reliable digits + last uncertain digit)which are directly obtained from aparticular measurement are called

significant figures.

Significant figures

Mass = 6.11 g3 significant figures

Speed = 67 km/h2 significant figures

Significant figures

Time = 12.76 s4 significant figures

Length = 1.8 cm2 significant figures

Rules for counting significant figures

1All non-zero digits are significant.

Number16 35.66438

Significant figures2 34

2Zeros between non-zero digits are significant.

Rules for counting significant figures

Number205

3008 60.005

Significant figures3 45

Rules for counting significant figures

3Terminal zeros in a number without decimal are not significant unless specified by a least count.

Number400 3050

(20 1) s

Significant figures1 32

Rules for counting significant figures

4Terminal zeros that are also to the right of a

decimal point in a number are significant.Number64.00 3.60

25.060

Significant figures4 35

Rules for counting significant figures5

If the number is less than 1, all zeroes before thefirst non-zero digit are not significant.

Number0.0064 0.0850

0.0002050

Significant figures2 34

6During conversion of units use powers of 10 to

avoid confusion.

Rules for counting significant figures

Number2.700 m

2.700 x cm2.700 x km

Significant figures444

Exact numbers• Exact numbers are either defined numbers or the

result of a count.• They have infinite number of significant figures

because they are reliable.By definition1 dozen = 12

objects 1 hour = 60 minute1 inch = 2.54 cm

By counting45 students

5 apples6 faces of

cube

Rules for rounding off a measurement

1If the digit to be dropped is less than 5, then the

preceding digit is left unchanged.Number64.62 3.651546.3

Round off up to 3 digits64.6 3.65546

2If the digit to be dropped is more than 5, then the

preceding digit is raised by one.Number3.47993.46683.7

Round off up to 3 digits3.48 93.5684

Rules for rounding off a measurement

3If the digit to be dropped is 5 followed by digits otherthan zero, then the preceding digit is raised by one.

Number62.3549.6552589.51

Round off up to 3 digits62.4 9.66590

Rules for rounding off a measurement

4If the digit to be dropped is 5 followed by zero or

nothing, the last remaining digit is increased by 1 if it is odd, but left as it is if even.

Number53.3509.455782.5

Round off up to 3 digits53.4 9.46782

Rules for rounding off a measurement

Significant figures in calculations

Addition & subtractionThe final result would round to the same decimal

place as the least precise number.Example:• 13.2 + 34.654 + 59.53 = 107.384 = 107.4• 19 – 1.567 - 14.6 = 2.833 = 3

Significant figures in calculations

Multiplication & divisionThe final result would round to the same numberof significant digits as the least accurate number.

Example:• 1.5 x 3.67 x 2.986 = 16.4379 = 16• 6.579/4.56 = 1.508 = 1.51

Errors in measurement

Difference between the actual value ofa quantity and the value obtained by a

measurement is called an error.

Error =

Types of errors

Systematic errorsGross errors

Random errors

1. Systematic errors

• These errors are arise due to flaws in experimental system.

• The system involves observer, measuring instrument and the environment.

• These errors are eliminated by detecting the source of the error.

Types of systematic errors

Personal errorsInstrumental errors

Environmental errors

a. Personal errorsThese errors are arise due to faulty proceduresadopted by the person making measurements.

Parallax error

b. Instrumental errorsThese errors are arise due to faulty

construction of instruments.

Zero error

c. Environmental errors

These errors are caused by external conditions like

pressure, temperature, magnetic field, wind etc. Following are the steps that one must follow in order to eliminate the environmental errors:a. Try to maintain the temperature and humidity of the

laboratory constant by making some arrangements.b. Ensure that there should not be any external magnetic

or electric field around the instrument.

Advanced experimental setups

2. Gross errorsThese errors are caused by mistake in using instruments, recording data and calculating

results. Example:a. A person may read a pressure gauge indicating 1.01 Pa

as 1.10 Pa.b. By mistake a person make use of an ordinary electronic

scale having poor sensitivity to measure very low masses.Careful reading and recording of the data can reduce

the gross errors to a great extent.

3. Random errors

• These errors are due to unknown causes and are sometimes termed as chance errors.

• Due to unknown causes, they cannot be eliminated.

• They can only be reduced and the error can be estimated by using some statistical operations.

Error analysisFor example, suppose you measure the oscillation period of a pendulum with a stopwatch five times.

Trial no ( i ) 1 2 3 4 5Measured value ( ) 3.9 3.5 3.6 3.7 3.5

Mean valueThe average of all the five readings gives the most probable value for time period.

=

= =

= 3.64 = 3.6

Absolute errorThe magnitude of the difference between mean value and each individual value is called absolute error.

=

3.9 3.5 3.6 3.7 3.50.3 0.1 0 0.1 0.1

The absolute error in each individual reading:

Mean absolute errorThe arithmetic mean of all the absolute errors is called mean absolute error.

=

= =

= 0.12 = 0.1

Reporting of result• The most common way adopted by scientist and

engineers to report a result is:

Result = best estimate error

• It represent a range of values and from that we expect a true value fall within.

• Thus, the period of oscillation is likely to be within (3.6 0.1) s.

Relative errorThe relative error is defined as the ratio of the mean absolute error to the mean value.

=

/ = = 0.0277

/ = 0.028

Percentage errorThe relative error multiplied by 100 is called as percentage error.

percentage   error  =  relative   error   x  100

percentage error = 0.028 x 100 percentage error = 2.8

Least count errorLeast count error is the error associated with the resolution of the instrument.

• The least count error of any instrument is equal to its resolution.

• Thus, the length of pen is likely to be within (4.7 0.1) cm.

Combination of errors

• Let be absolute error in measurement of • Let be absolute error in measurement of • Let be absolute error in measurement of

In different mathematical operations like addition, subtraction, multiplication and

division the errors are combined according to some rules.

=

∆ X=∆ A+∆ B

= +

=

W hen X=An

=

=

Estimation

Estimation is a rough calculation to find an approximate value of

something that is useful for some purpose.

Estimate the number of flats in Dubai city

Estimate the volume of water stored in a dam

Order of magnitude

The approximate size ofsomething expressed in powers

of 10 is called orderof magnitude.

To get an approximate idea of the number, one may round the coefficient a to 1 if it is less than or equal to 5 and to 10 if it is greater than 5.Examples:• Mass of electron = 9.1 x kg kg kg• Mass of observable universe = 1.59 x kg kg kg

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