Measurement

MEASUREMENTRaveendra Tudangil

OVERVIEW• Importance of measurement• Key definitions• Requirements of a good physical unit• Classification of physical quantities• System of units• Need for SI system• Fundamental SI units• Supplementary SI units• Derived SI units with special names• Rules for the use of SI units• Multiples and Submultiples

OVERVIEW• Dimension• Uses of dimensions• Order of magnitude• Significant figures• Rules for significant figures• Rules for Approximation• Mistakes• Errors• Methods to minimise the errors• Error estimation

IMPORTANCE OF MEASUREMENT

• Measurement is the basics need in any research work in science.• Exact, careful and precise work of measurement leads to the discovery of very

important laws.• Measurement plays an important role in the market. Preparation of bills, money

transaction related to it etc. cannot be completed without measurement.• The quality of measurement depends upon the instrument used for the purpose. • Important physical data such as, specific heat of substances, latent heat, value of

acceleration due to gravity, specific resistances of different metals, different parameters of transistors etc. are the result of elaborate and careful experimental work based on precise measurements.

KEY DEFINITIONS

• A quantity that can be measured is called physical quantity.• Length, Mass, Speed, Velocity, Time, etc.• Hate, Affection, Love, etc. cannot be measured.

• Measurement of a physical quantity is its comparison with some accepted standard which is called the unit of that physical quantity.• Any measurement is thus a sort of comparison, comparison of the physical

quantity with the unit of that physical quantity.• A standard used for measurement of a physical quantity is called the unit of

that physical quantity.

REQUIREMENTS OF A GOOD PHYSICAL UNIT

• A good physical unit must be of a suitable size and well defined.• A good physical unit should be invariable i.e. its magnitude should not vary

with time and place.• A good physical unit should be easily reproducible. Thus it should be very

easy to copy the unit so that it can be readily available.• It should be universally accepted.

Thus in short “A unit is, that standard which contains a definite, durable, stable and easily reproducible amount of the same kind of physical quantity and with which other similar quantities are compared”.

CLASSIFICATION OF PHYSICAL QUANTITIES

Fundamental Physical Quantities• Do not depend on any other physical quantity for

their measurement• E.g. Mass, Time• Their units are called fundamental units

Derived Physical Quantities• Depend of other physical quantities for their

measurement• E.g. Density, Area• Their units are called derived units

SYSTEM OF UNITS

• A system of units consists of fundamental units and derived units.• Formerly different system of units were being used in different countries.

System of Units

Origin Unit of Length

Unit of Mass Unit of Time

CGS System (Gaussian System)

French Academy of Sciences

centimeter gram second

FPS System (British System)

Great Britain foot pound second

MKS System (Metric System)

French Academy of Sciences

meter kilogram second

NEED FOR SI SYSTEM• Prior to SI system, different systems of units were used in different countries.• Therefore it was difficult for the scientists all over the world to exchange scientific information

between them.• This difficulty was overcome by accepting SI system by international agreement.• The SI system has totally removed this communication gap.• The laboratories all over the world are now using this common language of SI units and their

symbols.• Due to common language of SI units, the comparison between the scientific data from

different laboratories all over the world could be done easily and quickly.• This has accelerated the research work undertaken by the scientists, engineers and

technologists all over the world.• SI system has really become basic means for the vast and fast progress in the scientific world.

FUNDAMENTAL SI UNITS

Fundamental Quantity SI base unit SymbolLength meter mMass kilogram kgTime second sElectric Current ampere ALuminous Intensity candela cdTemperature kelvin KAmount of Substance mole mol

SUPPLEMENTARY SI UNITS

Fundamental Quantity SI base unit SymbolPlane Angle radian radSolid Angle steradian sr

DERIVED SI UNITS WITH SPECIAL NAMES

Fundamental Quantity SI base unit SymbolForce newton NWork, Energy joule JPower watt WElectric Charge coulomb CElectric Potential volt VFrequency hertz HzMagnetic Flux weber WbMagnetic Induction tesla T

RULES FOR THE USE OF SI UNITS

• Full names of the units are not written with a capital initial letter.• metre not Metre• kilogram not Kilogram• newton not Newton

• For a unit named after a person, the symbol has a capital initial letter. Symbols of other units are not written with capital initial letters. • newton N not n• metre m not M

RULES FOR THE USE OF SI UNITS (CONTD.)

• Symbols are not written in plural form.• 10 m not 10 ms• 30 kg not 30 kgs

• Full stops or other punctuation marks are not written after the symbols of units.• 10 m not 10 ms.• 30 kg not 30 kgs.

MULTIPLES AND SUBMULTIPLESPrefix Symbol Multiplication Factortera T 1012

giga G 109

mega M 106

kilo k 103

hecto h 102

deca da 101

deci d 10-1

centi c 10-2

milli m 10-3

micro 10-6

nano n 10-9

pico p 10-12

DIMENSION• Dimensions of a physical quantity can be defined as the power to which the fundamental units

are raised to obtain the unit of that physical quantity.• Each fundamental unit is represented by a capital letter.

• Length L• Mass M• Time T• Electric Current I• Luminous Intensity J• Temperature

• Amount of Substance N• The dimensions are expressed using square brackets

• [Length]=[M0 L1 T0]

SOLVE• Find dimensions of

• Speed• Velocity• Acceleration• Momentum• Force• Work• Power• Pressure• Density• Angle• Impulse• torque

USES OF DIMENSIONS

• To verify the truth of the given physical equation according to the principle of homogeneous dimensions.

• To determine the conversion factor between the units of the same physical quantity in different systems.

• To derive the relation between different physical quantities.

ORDER OF MAGNITUDE

• Charge on electron e=0.000,000,000,000,000,00016 C• Radius of the earth R=6400,000 m• Velocity of light c=300,000,000 m/s

• Difficult to calculation!

• Charge on electron e=1.6x10-19 C• Radius of the earth R=6.4x106 m• Velocity of light c=3x108 m/s

very small or very large

Compact Representatio

n

Easy Calculation

ORDER OF MAGNITUDE(CONTD.)

• The order of magnitude of a number is thus defined as the power of ten which is nearest to that number.

• The order of magnitude of • Charge on electron e=10-19

• Radius of the earth R=106

• Velocity of light c=108 • Positive index of ten indicates that the given quantity is large.• Negative index of ten indicates that the given quantity is small.• All the mathematical operations such as multiplication, division, addition and subtraction

can be smoothly carried out using the notation of the order of magnitude.


• 83200 x 111 =8.32 x 104 x 1.11 x 102

=8.32 x 1.11 x 106

=9.235 x 106

• = = x 10-1 = 1.830 x 10-1

• = = x 1010 = 0.5524 x 1010

= 5.524 x 109


• 8.32 X 105 + 3.28 X 106 = 8.32 X 105 + 32.8 X 105

= (8.32 + 32.8) X 105

= 41.12 X 105 = 4.112 X 106

• 9.12 X 106 - 2.8 X 105 = 9.12 X 106 - 0.28 X 106

= (9.12 - 0.28) X 106

= 8.84 X 106

SIGNIFICANT FIGURES

• The accuracy of a physical quantity is properly indicated by the number of figures which are used in the expression of a numeral measure. Conventionally only those figures retained which are reasonably trustworthy.

• The number of figures to specify a certain measurement are called significant figures. Though the last figure of the measurement is always doubtful, it is included in the number of significant figures.

• It is a general tendency to retain too many figures in the calculations, but it simply adds to the arithmetic labour which leads to fictitious precise result.

RULES FOR SIGNIFICANT FIGURES

• All non zero digits are significant.• All zeros occurring between two non zero digits are significant.• All zeros to the right of the decimal point and to the left of a non zero digit are

significant.• All zeros to the right of the last non zero digit are not significant. But all zeros to

the right of the last non zero digit are significant provided they come from an actual measurement.

• The greater the number of significant figures in a measurement, the smaller is the percentage error.

SOLVE

Quantity Number of Significant Figures490.0510.7600.08903.0281.0740

SOLVE

Quantity Number of Significant Figures49 20.051 30.760 20.0890 33.028 41.0740 5

RULES FOR APPROXIMATION

• A number is rounded off to any desired number of the significant figures by dropping one or more digits to the right.

• In dropping figures that are not significant, the last figure retained should be unchanged if the first figure dropped is less than 5.

• It should be increased by 1 if the last figure dropped is greater than 5.• If the last figure is 5, the preceding digit should be unchanged, if it is an even number

but increased by 1, if it is an odd number.• 3.447 3.48• 5.385 5.38• 4.8705 4.870

WHY TO CONSIDER ERRORS?

Find Error Cause

Extend of Error

Extend of UncertaintyExtend of Accuracy

Trustworthiness

MISTAKES

• Mistakes are faults caused by the carelessness of an untrained experimenter. Hence, mistakes can be avoided by careful and properly trained experimenter.

• Mistakes are caused by• Lack of skill• Faulty observation• Wrong readings• Incorrect mathematical manipulation

• Hence mistakes can be totally avoided by

• Skill• Correct observations• Correct readings• Correct mathematical

manipulation

ERROR

• The errors are the short comings which can occur even in most careful observations.

• Errors cannot be completely eliminated but they can be minimised by using proper methods.

Errors

Gross Errors Systematic Errors

Instrumental Errors

Environmental Errors

Observational Errors

Random Errors

METHODS TO MINIMISE THE ERRORS

• A large magnitude of the physical quantity should be taken for measurement.• While determining the period of a simple pendulum, the time taken for a definite number of

oscillations is measured in seconds then by dividing the time taken, by the number of oscillations, the periodic time is calculated. If the number of oscillations for which the time is measured is large, the error involved in the determination of the periodic time is small.

• The least count of the instrument used for the measurement should be as small as possible.

• In any measurement, a large number of readings should be taken and their average value should be determined. Such an average value is the best estimate of the magnitude of the physical quantity to be measured.

ERROR ESTIMATION

• Let A1, A2, A3, …, An be the values of ‘n’ number of observations in a measurement.

• Average Value (Am)• Absolute Error• Average Absolute Error• Relative Error• Percentage Error

Am=

ERROR ESTIMATION



Am-A1=A1 : Absolute error in the measurement of A1Am-A2=A2 : Absolute error in the measurement of A2

-----------------------------------------------------------------------------------------------------

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Am-An=An : Absolute error in the measurement of An

The difference between the average value and each measured value is called the absolute error in that

measurement.

ERROR ESTIMATION



Am=

The average of all the absolute errors is called the average absolute error (Am)

ERROR ESTIMATION



The ratio of the average absolute error to the average value is called the relative error in the

measurement.

Relative error =

ERROR ESTIMATION



The relative error expressed as a percentage is called the percentage error.

Percentage error =

RULES TO DETERMINE THE PERCENTAGE ERROR IN A COMPOUND QUANTITY

• If the error in is ,then the percentage error is X 100• If the error in is ,then the percentage error in is X 100• If the error in is , and the error in is then,

• The percentage error in = X 100• The percentage error in = X 100

SOLVE

• An experiment is conducted to measure the acceleration due to gravity. Following is the set of values obtained in the experiment.• 989.5 cm/s2

• 984.7 cm/s2

• 977.7 cm/s2

• 976.5 cm/s2

• 982.4 cm/s2

• Find the relative and percentage error in the measurement.

THANK YOU

Measurement

Science

Transcript of Measurement