Wednesday 3 February 2021

Chapter 2 Units And Measurement


Chapter 2 Units And Measurement




Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a

measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit.

Although the number of physical quantities appears to he very large, we need only a limited number of units for

expressing all the physical quantities, since they are inter-related with one another. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the

derived quantities are called derived units. A complete set of these units, both the base unite and derived units, is known as the system of units.




In earlier time scientists of different countries were using different systems of units for measurement. Three such

systems, the CGS, the FPS (or British) system and the MKS system were in use extensively till recently.


The base units for length, mass and time in these systems were as follows :

In CGS system they were centimetre, gram and second respectively.


In FPS system they were foot, pound and second respectively.


In MKS system they were metre, kilogram and second respectively.


The system of units which is at present internationally accepted for measurement is the Systéme Mtermationale d’ Unites (French for International System of Units),abbreviated as SI. The SI, with standard scheme of symbols,untts and abbreviations, was developed and recommended by General Conference on Weights and Measures in 1971 for


international usage in scientific, technical,industrial and commercial work. Because SI units used decimal system, conversions within the system are quite simple and convenient. We

shall follow the SI unita in this book.


In SI, there are seven base units as given in Table 2.1. Besides the seven base units, there are two more units that are defined for (a) plane angle d@as the ratio of length of arc ds to the radius rand (b) solid angle dQ as the ratio of the intercepted area dA of the spherical surface,described about the apex O as the centre, to the square of its radius r, as shown in Fig. 2.1a)

and {b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the sold angle is steradian with the symbol sr. Both these are dimensionless quantities.


Note that when mole is used, the elementary entitiea must be specified. These entities may be atoms, molecules, ions, electrons,other particles or speciiled groups of such particles.


We employ units for some physical quantities that can be derived from the seven base units (Appendix A 6). Some derived units in terms of the SI base units are given in (Appendix A 6.1).

Some SI derived untts are given special names (Appendix A 6.2) and some derived SI units mak:use of these units with special names and the seven base units (Appendix A 6.3). These are given in Appendix A 6.2 and A6.3 for your ready

Teference. Other units retained for general use are given in Table 2.2.


Common SI prefixes and symbols for multiples and sub-multiples are given in Appendix A2.General guidelines for using symbols for physical quantities, chemical elements and nuclides are

given in Appendix A7 and those for SI units and some other units are given in Appendix A8s for your guidance and ready reference.



You are already familiar with some direct methods for the measurement of length. For exampk, a metre scale is used for lengths from 10*m to 10?

m. A vernier callipers is used for lengths to an accuracy of 10*m. A screw gauge and a spherometer can be used to measure lengths as less as to 10m. To measure lengths beyand these Tanges, we make use of some special indirect



2.3.1 Measurement of Large Distances

Large distances such as the distance ofa planet or a star from the earth cannot be measured directly with a metre acale. An important method in such cases is the parallax method.


When you hold a pencil in front of you against some specific point on the background (a wall)and look at the pencil first through your left eye

A (closing the right eye) and then look at the pencil through your right eye B (closing the left eye), you would notice that the position of the pencil seems to change with respect to the paint

on the wall. This is called parallax. The distance between the two points of observation is called the basis. In this example, the basis is the distance between the eyes.


To measure the distance D of a far away

planet 5 by the parallax method, we observe it from two different positions (observatories) Aand Bonthe Earth, separated by distance AB = b

at the same time as shown in Fig. 2.2. We measure the angle between the two directions along which the planet is viewed at these two points. The ZASB in Fig. 2.2 represented by symbol 6 is called the parallax angle or

perallactic angle.


As the planet is very far away, p< 1. and therefore, g@ is very small. Then we

approximately take AB as an arc of length bofa circle with centre at S and the diatance D as 

Having determined D, we can employ a similar method to determine the size or angular diameter

of the planet. If d is the diameter of the planct and « the angular size of the planet (the angle subtended by d at the earth), we have a=d/D (2.2)

The angle a can be measured from the same location on the carth. It is the angle between the two directions when two diametrically opposite points ofthe planct are viewed through the telescope. Since Dis known, the diameter d

of the planet can be determined using Eq. (2.2).


Example 2.1 Calculate the angle o

(a) 1° (degree) (h) 1’ Gninute of arc or arcmin)and (c) 1*(second of arc or arc second) in radians. Use 360°=2n rad, 1°=60’ and 1’=60” Answer 

(a) We have 360°= 2n rad 1° = (x /180) rad = 1.745x 107 rad

 (b} 1° = 60’ = 1.745x10? rad 1’= 2.908107 rad = 2.91x10* rad 

(c) 1’= G0" = 2.908x10" rad 1°= 4.847x10* rad = 4.85x10°rad

the distance of a nearby tower from him.


He stands at a point A in front of the tower C and spots a very distant object O in line with AC. He then walks perpendicular to AC up to B, a distance of 100 m, and looks at O and C again. Since O is very distant,the direction BO is practically the same as AG; but he finds the Hne of sight of C shifted

from the original line of sight by an angle 6 = 40° (8 is known as ‘parallax’) catimate the distance of the tower C from his original position A.Answer We have, parallax angle 6 = 40° From Fig. 2.3, AB = AC tan 0 AC = AB/tan@ = 100 m/tan 40° = 100 m/0.8391 = 119m 4

Example 2.3 The moon is observed from

two diametrically oppostte points A and B on Earth. The angle @ subtended at the

moon by the two directions of observation is 1°54’. Given the diameter of the Earth to be about 1.276 x 10’ m, compute the distance of the moon from the Earth.Answer We have 6 = 1°54 = 114

=(114x60) x(4.85x10*) rad = 332x107 rad,

since 1’ = 4.85x10°rad.Also b= AB =1.276x10"m Hence from Eq. (2.1), we have the earth-moon distance,D=b/0

Z 1.276x10'3.32107 =3.84x10%m

Example 2.4 The Sum’s angular diameter

is measured to be 1920”. The distance Dof the Sun from the Earth is 1.496 x 10" m.What is the diameter of the Sun ?

Answer Sun's angular diameter a&

= 1920"

= 1920% 4.85% 107° rad

= 9.31x10~ rad

Sun's diameter


=(9.3n10- x(1 496x210! 1 } m

=1.39x10°m 4


2.3.2 Estimation of Very Small Distances:Size of a Molecule To measure a very small size like that of a

molecule (10° m to 10°'° m), we have to adopt special methods. We cannot use a screw gauge or similar instruments. Even a microscope has certain limitations. An optical microscope uses visible light to ‘look’ at the system under investigation. As light has wave like features,the resolution to which an optical microscope can be used is the wavelength of light (A detailed

explanation can be found in the Clasa XII Physica textbook). For visible light the range of wavelengths is from about 4000 A to 7000 A (1 angstrom = 1 A= 10°° m). Hence an optical microscope cannot resolve particles with sizes smaller than this. Instead of visible light, we can use an electron beam. Electron beams can be focussed by properly designed electric and magnetic fields. The resolution of such an electron microscope is limited finally by the fact that electrons can also behave as waves | (You will learn more about this in class XII). The wavelength of an electron can be as small as a

fraction of an angstrom. Such electron

microscopes with a resolution of 0.6 Ahave been built. They can almost resolve atoms and molecules in a material. In recent times,tumnelling microscopy has been developed in

which again the limit of resolution is better than an angstrom. It ia possible to estimate the sizes of molecules.


Asimple method for estimating the molecular size of oleic acid is given below. Oleic acid is a soapy liquid with large molecular size of the order of 10°m.


The idea is to first form mono-molecular layer of oleic acid on water surface.

We dissolve 1 cm of oleic acid in alcohol to make a solution of 20 cm®. Then we take 1 cm® of this solution and dilute it to 20 cm*, using alcohol. So, the concentration of the solution is

equal to (stag |e" of oleic acid/cem® of

20x20 solution. Next we lightly sprinkle some lycopodium powder on the surface of water ina large trough and we put one drop of this solution in the water. The oleic acid drop spreads into a thin, large and roughly circular film of molecular thickness on water surface. Then, we quickly measure the diameter of the thin film to get its area A, Suppose we have dropped n drops in the water. Initially, we determine the approximate volume of each drop (Vcm’}.Volume of n drops of solution =nVcm?Amount of oleic acid in this solution anv (—— Jem

~ (20x20 This solution of oleic acid spreads very fast on the surface of water and forms a very thin layer of thickness £. If this spreads to form a film of area A cm?, then the thickness of the film t Volume of the film

Area of the film or" 90x20A~ 2.3)


If we assunne that the film has mono-molecular thickness, then this becomes the aize or diameter of a molecule of oleic acid. The value of this thickness comes out to be of the order of 10° m.

Example 2.5 If the size of a nucleus {in

the range of 10 to 104m) is scaled up

to the tip of a sharp pin, what roughly is the size of an atom ? Assume tip of the pin to be in the range 10*m to 10“m.

Answer The size of a nucleus is in the range of 10% m and 10% m. The tip of a aharp pin is taken to be in the range of 10° m and 10* m.Tims we are scaling up by a factor of 10". An atom roughly of size 10°” m will be scaled up toa

size of 1 m. Thus @ nucleus in an atom is as small in size as the tip ofa sharp pin placed at the centre of a sphere of radius about a metre long. 4


2.3.3 Range of Lengths

The sizes of the objects we come across in the universe vary over a very wide range. Theae may vary from the size of the order of 10-“ m of the tiny nucleus of an atom to the size of the order

of 107° m of the extent of the observable universe.Table 2.6 gives the range and order of lengths and aizes of aome of these objects.We also use certain special length units for

short and large lengths. These are

1 fermi =lf=10"m

1 angstrom =1A=10°m

l astronomical unit = 1 AU [average distance

of the Sun from the Earth)

= 1.496 x 10" m


1 light year = 1 ly= 9.46 x 10% m (distance that light travels with velocity of 3x 105ms" in 1 year}1 parsec = 3.08 x 10'° m (Parsec is the distance at which average radius of earth's orbit

subtends an angle of 1 arc second)



Mass is a basic property of matter. It doce not depend on the temperature, pressure or location of the object in space. The SI unit of mass is

Kdlogram (kg). The prototypes of the International standard kilogram supplied by the International Bureau of Weights and Measures (BIPM) are available in many other laboratories of different

countries. In India, this is available at the National Physical Laboratory (NPL}, New Delhi.


While dealing with atoms and molecules, the kdlogram ts an inconvenient unit. In this case,there is an important standard unit of mass,called the unified atomic mase unit (u), which has been established for expressing the masa

of atoms as 1 unified atomic mass unit = 1u = (1/12) of the mass ofan atom of carbon- 12 isotope (3c) inchiding the mass of electrons = 1.66x 10” kg


Mass of commonly available objects can be determined by a common balance like the one used in a grocery shop. Large massea in the universe like plancts, stars, etc., based on Newton's law of gravitation can be measured by

using gravitational method (See Chapter 8). For measurement of small masses of atomic/sub-atomic particles etc., we make use of mass spectrograph in which radius of the trajectory is proportional to the mass ofa charged particle

moving in uniform electric and magnetic field.



2.4.1 Range of Masses

The masses of the objects, we come across in the universe, vary over a very wile range. These may vary from tiny mass of the order of 10 kg of an electron to the huge mass of about 10° kg of the known universe. Table 2.4 gives the range and order of the typical masses of various objects.



To measure any time interval we need a clock.We now use an stomic standard of time, which is based on the periodic vibrations produced in a cesium atom. This is the basis of the cesium

clock, sometimes called atomic clock, used in.the national standards. Such standards are available in many laboratories. In the cestim

atomic clock, the second is taken as the time needed for 9,192,631,770 vibrations of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133 atom. The vibrations of the cesium atom regulate the rate of this ceshim

atomic clock Just as the vibrations ofa balance wheel regulate an ordinary wristwatch or the vibrations of a small quartz crystal regulate a quartz wristwatch.


The cesium atomic clocks are very accurate.In principle they provide portable standard. The national standard of time interval ‘second’ as well aa the frequency fa maintained through four

cesium atomic clocks. Acestum atomic clock is used at the National Physical Laboratory (NPL},New Delhi to maintain the Indian standard of time.


In our country, the NPL has the responsibility of maintenance and improvement of physical standards, inchiding that of time, frequency, etc.

Note that the Indian Standard Tine (IST) is linked to this set of atomic clocks. The efficient cesium atomic clocks are ao accurate that they impart the uncertainty in time realisation as


+1x10-%, Le. 1 part in 10. This implies that the uncertainty gained over time by such a device is less than 1 part in 10"; they lose or gain no more than 3 j1s in one year. In view of the tremendous accuracy in time measurement,

the SI unit of length has been expressed in terms the path length light travels in certain interval of time (1/299, 792, 458 of a second) (Table 2.1).


The time interval of events that we come

across in the universe vary over a very wide range. Table 2.5 gives the range and order of some typical time intervals.


You may notice that there is an interesting coincidence between the numbers appearing in Tables 2.3 and 2.5. Note that the ratio of the longest and shortest lengths of objects in our

universe is about 10*!. Interestingly enough,the ratio of the longest and shortest time intervals associated with the events and objects in our universe is also about 10“. This number,

10*! comes up again in Table 2.4, which lists typical masses of objects. The ratio of the largest and smallest masses of the objects in our universe is about (10*!)?. Ie this a curious coincidence between these large numbers purely accidental ?




Measurement is the foundation of all experimental acience and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Every calculated quantity which is based on measured values,also has an ertor. We shall distinguish between

two terms: accuracy and precision. The

accuracy ofa measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured.


The accuracy in measurement may depend on several factors, inchiding the limit or the reschition ofthe measuring instrument. For exsanple, suppose

the true value ofa certain length is near 3.678 cm.In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to

be 3.5 cm, while in another experiment using a Measuring device of greater resolution, say 0.01 cm,the length is determined to be 3.38 cm. The first

measurement has more accuracy (because it is closer 

to the true value) but less precision {its resolution is only 0.1 cm), while the second measurement is less accurate but more preciae. Thus every measurement is approximate due to errors in measurement. In general, the errors in measurement can be broadly classified as

(a) systematic errors and

(b) random errors.


Systematic errors

The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are :


(a) Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instrument, Zero error in the instrument, etc. For example, the temperature graduations of a thermometer may be inadequately calibrated (it may read 104 °C at the boiling point of water at STP whereas it should read 100 °C); in a vernier

callipers the zero mark of vernier scale may not coincide with the zero mark of the main scale, or aimply an ordinary metre scale may be worn off at one end.


(b) Imperfection in experimental technique or procedure To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual valine of the body temperature. Other external conditions (such as changes in temperature,humidity, wind velocity, etc.) during the experiment may systematically affect the measurement.


(c) Personal errors that arise due to an

individual's bias, lack of proper setting of the apparatus or individual's carelessness in taking observations without observing proper precautions, etc. For example, ifyou,by habit, always hold your head a bit too far to the right while reading the position of a

needle on the scale, you will introduce an error due to parallax.


Systematic errors can be minimised by

improving experimental techniques, selecting better instruments and removing personal bias as far as possible. For a given set-up, these

errors may be estimated to a certain extent and the necessary corrections may be applied to the readings.


Random errors

The random errors are those errors, which occur irregularly and hence are random with respect closer to the true value) but less precision {its resolution is only 0.1 cm), while the

second measurement is less accurate but

more preciae. Thus every measurement is

approximate due to errors in measurement. In general, the errors in measurement can be broadly classified as (a) systematic errors and

(b) random errors.


Systematic errors

The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are :


(a) Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instrument, Zero error in the instrument, etc. For example, the temperature graduations of a thermometer may be inadequately calibrated (it may read 104 °C at the boiling point of water at STP

whereas it should read 100 °C); in a vernier callipers the zero mark of vernier scale may not coincide with the zero mark of the main scale, or aimply an ordinary metre scale may be worn off at one end.


(b) Imperfection in experimental technique or procedure To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual valine of the body temperature. Other external conditions (such as changes in temperature,humidity, wind velocity, etc.) during the experiment may systematically affect the measurement.


(c) Personal errors that arise due to an

individual's bias, lack of proper setting of the apparatus or individual's carelessness in taking observations without observing proper precautions, etc. For example, ifyou,by habit, always hold your head a bit too far to the right while reading the position of a

needle on the scale, you will introduce an error due to parallax.


Systematic errors can be minimised by

improving experimental techniques, selecting better instruments and removing personal bias as far as possible. For a given set-up, these

errors may be estimated to a certain extent and the necessary corrections may be applied to the readings.


Random errors

The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random

and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical

vibrations of experimental set-ups, etc), personal (unbiased) errora by the observer taking Teadings, etc. For example, when the same person repeats the same observation, it is very

likely that he may get different readings everytime.


Least count error

The smallest value that can be measured by the measuring instrument is called its least count.All the readings or measured values are good only

up to this value.


The least count error is the error

associated with the resolution of the instrument.For example, a vernier callipers has the least count as 0.01 cm; a spherometer may have a

least count of 0.001 cm. Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors. If we use a metre scale for measurement of length, it may have

graduations at 1 mm division scale spacing or interval.


Using instruments of higher precision,

improving experimental techniques, etc., we can Teduce the least count error. Repeating the observations several times and taking the arithmetic mean of all the observations, the mean value would be very close to the true value

of the measured quantity.


2.6.1 Absolute Error, Relative Error and

Percentage Error

(a) Suppose the values obtained in several measurements are a,, A, a,...., 4, The arithmetic mean of these values is taken as the best possible value of the quantity under the given conditions of measurement as :pean = (A+, +2,+..40,) | R (2.4)Or,, iH Aynccrr = Dy [1 (2.5)isl This is because, as explained earlier, it is reasonable to suppose that individual measurements are as likely to overestimate as to underestimate the true value of the



The magnitude of the difference

between the individual measurement and

the true value of the quantity is called the absolute error of the measurement. This is denoted by |Aa|, In absence of any other method of knowing true value, we considered arithmatic mean as the true value. Then the errors in the individual measurement values

from the true value, are Aa, =4,-d,

AG, = Q,~ yay Aa,= Q, — Brean


The Aa calculated above may be positive in certain cases and negative in some other cases. But absolute error |Aal will always be positive.


(b) The arithmetic mean ofall the absohie errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by Aa...



Aq, = (lAa,I+lAa, I+lAa,I+...+ lAa I) /a

(2.6)=> tdalsn 2.7 tol If we do a single measurement, the value we get may be in the range a+ Aa,ie @=a@ lit Aa

or,Poecn ~ AX gaan, SX S Apeey, + APpoe

(2.8) This imples that any measurement of the physical quantity a ts likely to Ife between Gt Mem) and (a, - Aa).


(c) Instead of the absolute error, we often use the relative error or the percentage error


(a). The relative error is the ratio of the mean absolute error 4a__ to the mean

value a. _ of the quantity measured.

Relative error = Aa, /a@,_, (2.9)

When the relative error is expressed in per cent, it is called the percentage error (5a).Thus, Percentage error

5A = (AG, / Ane) % 100% (2.10)

Let us now consider an example.

Example 2.6 Two clocks are being tested

against a standard clock located in a

national laboratory. At 12:00:00 noon by

the standard clock, the readings of the two clocks are ;Clock 1 Clock 2

Monday 12:00:05 10:15:06

Tuesday 12:01:15 10:14:59

Wedneaday 11:59:08 10:15:18

Thursday 12:01:50 10:15:07

Friday 11:59:15 10:14:53

Saturday 12:01:30 10:15:24

Sunday 12:01:19 10:15:11


If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer ?

Answer The range of variation over the seven days of observations is 162 s for clock 1, and 31s forclock 2. The average reading of clock 1 is much closer to the standard time than the average reading of clock 2. The important point

is that a clock’s zero error is not as significant for precision work as its variation, because a ‘zero-error’ can always be easily corrected.

Hence clock 2 is to be preferred toclock1. <Example 2.7 We measure the period of oscillation of a simple pendulum. In successive meastirements, the readings turn out to be 2.63 8, 2.56 8, 2.42 8, 2.719 and 2.80 8, Calculate the absolute errors,relative error or percentage error.Answer The mean period of oscillation of the pendulum

Te (2.634 2.564 2.42 +2.7142.80)s

- 5 13.12 =—s 5

= 2.624 8

=2.62 8


As the periods are measured to a resolution of0.01 s, all times are to the second decimal; it is proper to put thia mean period also to the

second decimal The errors in the measurements are

2.63 9-2.629= 0.0ls

2.56 8 -2.62 8 =-0.068

2.42 s - 2.628 =-0.208

2.718-2.628s= 0.099

2.80 8-2.628s= 0.188


Note that the errors have the same units as the quantity to be measured.The arithmetic mean of all the absolute errors (for arithmetic mean, we take only the magnitudes) is

AT... = [(0.01+ 0.06+0.20+0.09+0.18)8]/5

= 0.54 8/5



That meana, the period of oscillation of the simple pendulum is (2.62 + 0.11) s i.e. it lies between (2.62 + 0.11) s and (2.62 - 0.11) s or between 2.73 s and 2.51 9. As the arithmetic mean of all the absolute errors is 0.11 s, there

is already an error in the tenth of a second.Hence there is no point in giving the period to a hundredth. Amore correct way will be to write



Note that the last numeral 6 is unreliable, since it may be anything between 5 and 7. We indicate this by saying that the measurement has two

significant figures. In this case, the two significant figures are 2, which is reliable and 6, which has an error associated with it. You will learn more about the significant figures in

section 2.7.


For this example, the relative error or the percentage error is

, Ol

8a =— X100 = 19% |



2.6.2 Combination of Exrors

If we do an experiment involving several

measurements, we must know how the errors in all the measurements combine. For example,How will you measure the length of a Hine?

What a nalve question, at thia stage, you might say! But what if it ia not a straight line? Draw a zigzag Hine in your copy, or on the blackboard.

Well, not too difficult again. You might take a thread, place it along the line, open up the thread, and measure its length.


Now imagine that you want to measure the

length of a national highway, a river, the railway track between two stations, or the boundary between two states or two nations. If you take a atring of length 1 metre or 100 metre, keep it

along the Hine, shift tts position every time, the arithmetic of man-hours of labour and expenses on the project is not commer:surate with the outcome. Moreover, ators are bound to occur

in this enormous task. There is an interesting fact about this. France and Belgium share a conumon international boundary, whose length mentioned in the oficial documents of the two countries differs substantially!


Go one step beyond and imagine the

coastiine where land meets sea. Roads and rivers have fairly mild bends as compared te a coastline. Even so, all documents, including our school books, contain information on the length

of the coastline of Gujarat or Andhra Pradesh,or the common boundary between two atates,etc. Railway tickets come with the distance between stations printed on them. We have ‘milestones’ all along the roads indicating the

distances to various towns. So, how is it done?


One has to decide how much error one can

tolerate and optimise cost-ffectivenesa. If you waut smaller errors, it will involve high technology and high costs. Suffice it to say that it requires fairly advanced level of physics,

mathematics, engineering and technology. It belongs to the areas of fractals, which has lately become popular in theoretical physics. Even them one doean’t know how much to rely on

the figure that props up. as is clear from the story of France and Belgium. Incidentally, this story of the France-Belgium discrepancy appears on the first page of an advanced Physics book on the subject of fractale and chaos!density is obtained by deviding mass by the

volume of the substance. If we have errors in the measurement of mass and of the sizes or dimensions, we must know what the error will be in the density of the substance. To make such estimates, we should learn how errors combine

in various mathematical operations. For this,we use the following procedure.


(a} Error of a sum or a difference

Suppose two physical quantities A and B have Measured values A + AA, B + AB respectively where AA and AB are their absolute errors. We wish to find the error AZin the sum Z=A+B.We have by addition, Z+ AZ = (A 2 AA) + (B+ AB).

The maximum possible error in Z

AZ=AA+ AB For the difference Z= A- B, we have Z+AZ= (A+AA)—-(B+AB = (A-B)+ AA+ AB

Or, +AZ= +AA+AB


The maximum value of the error AZ is again AA + AB,

Hence the rule : When two quantities are

added or subtracted, the absolute error in the final result is the etm of the absolute errors in the individual quantities.Example 2.8 The temperatures of two bodies measured by a thermometer are t =20°C + 0.5 °C and ¢ = 50 °C +0.5°C.Calculate the temperature difference and the error theirin.

Answer ¢ = (+, = (50 °C#0.5 °C)— (20°C#0.5 °C)C=30°C#1°C 4


(b) Exror of a product or a quotient

Suppose Z= AB and the measured values of A and Bare A+ AAand B+ AB. Then


Dividing LHS by Zand RHS by AB we have,

12(AZ/Q = 1 + (AA/A) # (AB/B) * (AA/AN(AB/B).Since AA and AB are small, we shall ignore their product.Hence the maximum relative error A2/ Z= (AA/A) + (AB/B).You can easily verify that this is true for division also.Hence the rule : When two quantities are multiplied or divided, the relative error in the

reeult is the sum of the relative errors in the multipliers.


Example 2.9 The resistance R= V/Iwhere

V=(100 + 5)V and J=(10 + 0.2)A. Find the

percentage error in R.Answer The percentage error in Vis 5% and in

I itis 2%. The total error in R would therefore be 5% + 2% = 7%. 4

Example 2.10 Two resistors of resistances R, = 100+3 ohm and R, = 200 + 4 ohm are connected (a) in series, (b) in parallel. Find the equivalent resistance of the (a) series

combination, (b) parallel combination. Use for (a) the relation R =R, + R, and for (b)11,1 ‘AR’ _ AR, AR,—=— + — and ——_ =— 1 4.R’ R + R R®? R? + R,?

Answer (a) The equivalent resistance of series combination R=R,+R,= (100 + 3) ohm + (200 = 4) ohm = 300 + 7 ohm.

(b) The equivalent resistance of parallel combination Re RR, _ 200

R, +R, 3 = 66.7 ohm dott Then, from 5 RR,‘we get,AR’ AR | AR,RR? RZ, 2\4R 2\ AR,AR’= Re at R® — 2 (Rye HR Ye = 66.7 34 66.7 l 4 100 200 =1.8 Then, R’ = 66.7+1.8 ohm (Here, AR is expresed as 1.8 instead of 2 to keep in confirmity with the nules of significant figures.)



(c) Error in case of a measured quantity

raised to a power Suppose Z =A?,

Then,AZ/Z= (AA/A) + (AA/A) = 2 (AA/A).

Hence, the relative error in A? is two times the error in A.


In general, if Z= A? B/C

Then,AZ/Z= p (AA/A) + q(AB/B) +r (AC/Q.

Hence the rele : The relative crror in «

physical quantity ralsed to the power k is the k times the relative error in the individual quantity.


Example 2.11 Find the relative error in

Z, tf Z= ASB) / CD14,


Answer The relative error in Zis AZ/Z =

4(AA/A) +{1 /3) (AB/B) + (AC/O + (3/2) D/P.


Example 2.12 The period of oscillation of a simple pendulum is T= 2nJL/g.

Measured value of Lis 20.0 cm known to 1

mmm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of g?

Answer g=49°L/T* ‘t yp At ‘AT At

Here, T= — and AT=—. Therefore, ==".

The errora in both L and ¢ are the least count errors. Therefore,(Ag/g) = (AL/1) + 2{4T/T)0.1= ——+2| — |=0.027

= 300° (50 00


Thus, the percentage error in gis

100 (Ag/g) = 100(AL/E) + 2 x 100 (AT/T)

= 3% <



As discussed above, every measurement

involves errora. Thus, the result of

measurement should be reported in a way that indicates the precision of measurement.Normally, the reported result of measurement is a mumber that includes all digita in the number that are known reliably plus the first

digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures. If we say the period of oscillation of a simple

pendulum is 1.62 s, the digits 1 and 6 are reliable and certain, while the digit 2 is uncertain. Thus, the measured value has three significant figures. The length of an object Teported after measurement to be 287.5 cm has four significant figures, the digits 2, 8, 7 are certain while the digit 5 ia uncertain. Clearly,reporting the result of measurement that includes more digits than the significant digits 43 superfixous and also misleading since it would give a wrong idea about the precision of measurement.


The rules for determining the number of

significant figures can be understood from the following examples. Significant figures indicate,as already mentioned, the precision of measurement which depends on the least count of the measuring instrument. A choice of

change of different units does not change the number of significant digits or figures in a measurement. This tmportant remark makes most of the following observations clear:


(1) For example, the length 2.308 cm has four significant figures. But in different units, the same value can be written as 0.02308 m or 23.08 mmm or 23080 jum.


All these numbers have the same number of significant figures (digits 2, 3, 0, 8), namely four.This shows that the location of dectma! point is of no consequence in determining the number

of significant figures.


The example gives the following rules :

All the non-zero digits are significant.


All the zeros between two non-sero digits are significant, no matter where the decimal point is, if at all.


If the number is less than 1, the sero(s)on the right of decimal point but to the left of the first non-zero digit are not aignificant. [In 0.00 2308, the undertined zeroes are not significant].


The terminal or trailing zero(s) in «

number withont a decimal point are not

significant.[Thus 123m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s)being not aignificant.] However, you can also

see the next observation.


The tradling sero(s) in a number with a

decimal point are significant.[The numbers 3.500 or 0.06900 have four

significant figures each.]


(2) There can be some confusion regarding the trailing zero(s). Suppose a length is reported to be 4.700 m. It is evident that the zeroes here

are meant to convey the precision of

measurement and are, therefore, significant. [If these were not, tt would be superfluous to write them explicitly, the reported measurement

would have been simply 4.7 m]. Now suppose we change units, then

4.700 m = 470.0 om = 4700 mm = 0.004700 Im Since the last number has trafling zero(s) in a number with no decimal, we would conclude erroneously from observation (1} above that the

number has two significant figures, while in fact, it has four significant figures and a mere change of units cannot change the number of significant figures.


(3) To remove such ambiguities in

determining the number of significant

figures, the best way is to report every

measurement in scientific notation (in the power of 10). In this notation, every number is expressed as a x 10°, where a is a number between 1 and 10, and b is any positive or negative exponent {or power of 10. In order to

get an approximate idea of the number, we may round off the number a to 1 (for a <5) and to 10 (for 5<a¢16). Then the number can be expressed approximately as 10° in which the exponent (or power) b of 10 is called order of magnitude of the physical quantity. When only

an estimate is required, the quantity is of the order of 10". For example, the diameter of the earth (1.28x%10’m) js of the order of 10’m with the order of magnitude 7. The diameter of hydrogen atom (1.06 x107°m) is of the order of

10°"m, with the order of magnitude

—10. Thus, the diameter of the earth is 17 orders of magnitude larger than the hydrogen atom.


It is often customary to write the dectnal after the first digit. Now the confusion mentioned in

(a) above disappears :4,700 m = 4.700 x 107 cm = 4,700 x 107 mm = 4.700 x 10% km


The power of 10 is irrelevant to the

determination of significant figures. However, all zeroes appearing in the base number in the scientific notation are significant. Each number in this case has four significant figures.

Thus, in the scientific notation, no confision arises about the trailing zero(s) in the base mumber a. They are always significant.


(4) The scientific notation is ideal for reporting measurement. But if this is not adopted, we use the rules adopted in the preceding example :


Fora number greater than 1, without any

decimal, the trailing zero(s) are not



Fora number with a decimal, the trailing

sero(s) are significant.


(5) The digit O conventionally put on the left ofa decimal for a number less than 1 (like 0.1250)is never significant. However, the zeroes at the

end of such number are significant in a



(6) The multiplying or dividing factors which are neither rounded numbers nor numbers Tepresenting measured values are exact and have infinite number of significant digits. For example in rag or s = 2rr, the factor 2 is an

exact number and it can be written as 2.0, 2.00 or 2.0000 as required. Similarly, in Tet, nis an exact number.





2.7.1 Rules for Arithmetic Operations with significant Figures

The result ofa calculation involving approximate measured values of quantities (Le. values with Imitted number of significant figures) must reflect the uncertainties in the original measured values.


It cannot be more accurate than the original measured Values themselves on which the result is based. In general, the final result should not

have more significant figures than the original data from which it was obtained. Thus, ifmass of an object is measured to be, say, 4.237 g (four

significant figures) and its volume is measured to De 2.51 cm, then its density, by mere arithmetic division, is 1.68804780876 g/cm upto 11 decimal

places. It would be clearly absurd and irrelevant to recom the calculated value of density to sucha precision when the measurements on which the value is based, have much less precision. The

following rules for arithmetic operations with significant figures ensure that the final result of

a calculation is shown with the precision that is consistent with the precision of the input measured values :


(1) In multiplication or division, the final result should retain as many eignificant figures as are there in the original number with the least significant figures.


Thus, in the example above, density should be reported to three significant figures.

: 4.2378 23

Density = ————; 21.69 g cm


2.51 cm

Similarly, if the speed of light is given as 3.00 x 10° m s"* (three significant figures) and one year (Ly = 365.25 d) has 3.1557 x 10’ s (jive

significant figures), the ight year is 9.47 x 10% m (three significant figures).


(2) In addition or subtraction, the final result ehould retain as many decimal places as are there in the number with the least decimal



For example, the sum of the numbers

436.32 ¢, 227.2 ¢ and 0.301 g by mere arithmetic addition, is 663.621 g. But the least precise measurement (227.2 g) is correct to only one decimal place. The final result should, therefore,

be rounded off to 663.8 g.


Stinilarly, the difference in length can be expressed as :


0.307 m—0.304 m = 0.003 m =3 x 10° m.

Note that we should not use the rule (1)

applicable for multiplication and division and write 664 g as the result in the example of addition and 3.00 x 10° m in the example of subtraction. They do not convey the precision

of measurement properly. For addition and subtraction, the rule fs in terms of decimal places.


2.7.2 Rounding off the Uncertain Digits

The result of computation with approximate numbers, which contain more than one uncertain digit, should be rounded off. The rules for rounding off numbers to the appropriate significant figures are obvious in most cases. A

number 2.746 rounded off to three significant figures is 2.75, while the number 2.743 would be 2.74. The rule by convention is that the preceding digit is raised by 1 if the ineignificant digit to be dropped (the underlined digit in this case) is more than

G, and is left unchanged if the latter is less than &. But what if the number ts 2.745 in which the insignificant digit is 5. Here, the convention is that if the preceding digit is even, the insignificant digit is simply

dropped and, if it is odd, the preceding digit is raised by 1. Then, the mumber 2.745 rounded off to three significant figures becomes 2.74. On the other hand, the number 2.735 rounded off to three significant figures becomes


2.74 since the preceding digit is odd.

In any involved or complex multi-step

calculation, you should retain, in intermediate steps, one digit more than the significant digits and round off to proper significant figures at the

end of the calculation. Similarly, a number mown to be within many significant figures,such as in 2.99792458 x 10° m/s for the speed

of light in vacuum, is rounded off to an

approximate value 3 x 10° m/s . which is often employed in computations. Finally, remember that exact numbers that appear in formulae lke 2nin T= anf. have a large (infinite) number of significant figures. The value of « =3.1415926.... is known to a large number of

significant figures. You may take the value as 3.142 or 3.14 for a, with limited number of significant figures as required in specific cages.


Example 2.13 Each side of a cube is

measured to be 7.203 m. What are the

total surface area and the volume of the

Answer The number of significant figures in the measured length is 4. The calculated area and the vohime should therefore be rounded off to 4 significant figures.Surface area of the cube = 6(7.203)? m?=311.299254 m?

=311.3 m?*Volume of the cube = (7.203) m? =373.714754 m?= 373.7 m® <


Example 2.14 5.74 g of a substance

occupies 1.2 cm*. Express its density by

keeping the significant figures in view.

Answer There are 3 significant figures in the Measured mass whereas there are only 2 significant figures in the measured volume.Hence the density should be expressed to only 2 significant figures.oo BTA ly Density = 12 gem

=4.8 gem". <2.7.3 Rules for Determining the Uncertainty in the Reeultse of Arithmatic Calculations The rules for determining the uncertainty or error in the number/measured quantity in

arithmetic operations can be understood from the following examples.


(1) If the length and breadth of a thin

rectangular sheet are measured, using a metre scale as 16.2 cm and, 10.1 cm respectively, there are three significant figures in each measurement. It means that the length ! may be written as 1=16.220.1 cm = 16.20mz0.6 %,

Stnflarly, the breadth b may be written as b=10.1 20.1 em =10.lem21% Then, the error of the product of two (or more)

experimental values, using the combination of errors rule, will be

tb= 163.62 cm? + 1.6%= 163.62 + 2.6 cm?

This leada us to quote the final result as tb= 164+ 3 cm’ Here 3 cm? 1s the uncertainty or error in the estimation of area of rectangular sheet.


(2) If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also be valid to n significant figures.However, if data are subtracted, the number of significant figures can be reduced.


For example, 12.9 g—7.06 g, both specified to three significant figures, cannot properly be evaluated

as 5.64 g but only as 5.6 g, as uncertainties in subtraction or addition combine in a different fashion (smallest mimber of decimal places rather

than the number of significant figures in any of the number added or subtracted).


(3) The relative error of a value of number specified to significant figures depends not only on n but also on the number itecli.For example, the accuracy in measurement of mass 1.02 g is + 0.01 g whereas another measurement 9.89 g is also accurate to + 0.01 g.The relative error in 1.02 g is = (4 0.01/1.02) x 100 % =+1% Similarly, the relative error in 9.89 g is  = (40.01 /9.89) x 100 %



Finally, remember that intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.These should be justified by the data and then

the arithmetic operations may be carried out;otherwise rounding errora can build up. For example, the reciprocal of 9.58, calculated (after rounding off) to the same number of significant figures (three) is 0.104, but the reciprocal of

0.104 calculated to three significant figures is 9.62. However, ifwe had written 1 /9.58 = 0.1044 and then taken the reciprocal to three significant figures, we would have retrieved the original value of 9.58.


This example justifies the idea to retain one more extra digit (than the number of digits in the least precise measurement) in intermediate steps of the complex multi-step calculations in

order to avoid additional errors in the process of rounding off the numbers.



The nature of a physical quantity is described by its dimensions. All the physical quantities represented by derived units can be expressed

in terms of some combination of seven

fundamental or base quantities. We shall call these base quantities as the seven dimensions of the physical world, which are denoted with square brackets [ ]. Thus, length has the dimension [L], mass [M], time [T], electric current

[A], thermodynamic temperature [K], hminous intensity [cd], and amount of substance [mol].The dimensions of a physical quantity are the powers (or exponents) to which the base

quantities are raised to represent that

quantity. Note that using the square brackets [ ] round a quantity means that we are dealing with ‘the dimensions of’ the quantity.


In mechanics, all the physical quantities can be written in terms of the dimensions [L], [M]and [T]. For example, the volume occupied by an object is expreased as the product of length,breadth and height, or three lengths. Hence the dimensions of vohume are [L] x [LJ x [LJ = [L}* = [L4.

As the volume is independent of mass and time,it is said to poasess zero dimension in mass [M’],zero dimension in time [T°] and three dimensions in length.


Similarly, force, as the product of mass and acceleration, can be expreased as

Force = mass x acceleration = mass x (length) /(time)

The dimensions of force are [M] [L]/[T]? =[ML T*]. Thus, the force has one dimension in mass, one dimension in length, and -2 dimensions in time. The dimensions in all other base quantities are zero.


Note that in this type of representation, the magnitudes are not considered. It is the quality of the type of the physical quantity that enters.Thus, a change in velocity, initial velocity,average velocity, final velocity, and speed are all equivalent in this context. Since all these

quantities can be expressed as length/time,their dimensions are [L]/{T] or IL T*].




The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional JSormuta of the given physical quantity. For

example, the dimensional formula of the volume is [M° L? T°], and that of speed or velocity is [IM° LT]. Similarly, [M° LT] is the dimensional formula of acceleration and [M L* T°] that of

mass density.


An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity. Thus, the dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities. For example, the dimensional equations of volume [V], speed [vu], force [F] and

mass density [p] may be expressed as

[V] = IML? T°

[v) = PLT")

[Al = [ML T*

[pl = ML* TY


The dimensional equation can be obtained

from the equation representing the relations between the physical quantities. The dimensional formulae of a large number and wide variety of physical quantities, derived from

the equations representing the relationships among other physical quantities and expreased in terms of base quantities are given in Appendix 9 for your guidance and ready





The recognition of concepts of dimensions, which guide the description of physical behaviour is of basic importance as only those physical

quantities can be added or subtracted which have the s ame dimensions. A thorough understanding of dimenatonal analyais helps us in deducing certain relations among different physical quantities and checking the derivation,

accuracy and dimensional consistency or

homogeneity of various mathematical

expressions. When magnitudes of two or more physical quantities are multiplied, their units should be treated in the same manner as ordinary algebraic symbols. We can cancel identical units in the numerator and

denominator. The same is true for dimensions of a physical quantity. Similarly, physical quantities represented by symbols on both sides

ofa mathematical equation must have the same dimensions.


2.10.1 Checking the Dimensional

Consistency of Equations

The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. In other words, we can add or subtract similar physical quantities. This, velocity cannot be added to

force, or an electric current cannot be subtracted from the thermodynamic temperature. This simple principle called the principle of homogeneity of dimensions in an equation is extremely useful in checking the correctness of

an equation. If the dimensions of all the terms are not same, the equation is wrong. Hence, if we derive an expression for the length {or distance) of an object, regardless of the symbols

appearing in the original mathematical relation,when all the individual dimensions are simplified, the remaining dimension must be that of length. Similarty, if we derive an equation

of speed, the dimensions on both the sides of equation, when simplified, must be of length/ time, or [L T"}].


Dimensions are customarily used as a

preliminary test of the consistency of an equation, when there is some doubt about the correctness of the equation. However, the dimensional consistency does not guarantee correct equations. It is uncertain to the extent of dimensionless quantities or functions. The arguments of special functions, such as the trigonometric, logarithmic and exponential functions must be dimensionless. A pure number, ratio of similar physical quantities,such as angle as the ratio (length/length),

refractive Index as the ratio (speed of light in vacuum /sapeed of light in medium) etc., has no dimensions.


Now we can teat the dimensional consistency or homogeneity of the equation X=x, +0, 64+0/2)a

for the distance x travelled by a particle or body in time t which starts from the position x, with an initial velocity uv, at the t= 0 and has uniform

acceleration a along the direction of motion.


The dimensions of each term may be written as

bd = [LI

bx] = 4

ly, 2 =T) (1

=(1/2) a®) my T?] (P


As each term on the right hand side of this equation has the same dimension, namely that of length, which is same as the dimension of left hand side of the equation, hence this equation is a dimensionally correct equation.It may be noted that a test of consistency of

dimensions tells us no more and na less than a test of consistency of units, but has the advantage that we need not commit ourselves to a particular choice of units, and we need not worry about conversions among multiples and

sub-multiples of the units. It may be borne in mind that if an equation fafls this consistency teat, it ia proved wrong, but if it passes, it is

not proved right. Tin, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be

wrong,Example 2.15 Let us conskier an equation ia 2 mvo=-mghkh


where mis the mass of the body, uv tts

velocity, g js the acceleration due to

gravity and his the height. Cheek

whether this equation is dimensionally



Answer The dimensions of LHS are

[IM] [L T* = M] [LT

= (MI? T?]

The dimensions of RHS are

ML T*) 14 = ML T*]

= (MIT?


The dimensions of LHS and RHS are the same and hence the equation is dimensionally correct. <Example 2.16 The SI untt of energy 1s J= kgm’s*; that of speed vis ms“ and of acceleration ais ms%. Which of the formulae for kinetic energy (K) given below can you rule out on the basis of dimens{onal arguments (m stands for the mass of the bocty) :

(a) K = nm? v?

(b) K= (1/2)mv?

(c) K=ma

(d) K=(3/16}m7

(e) K=(1/2}mv?+ ma


Answer Every correct formula or equation must have the same dimensions on both sides of the equation. Also, only quantities with the same physical dimensions can be added or subtracted. The dimensions of the quantity on

the right side are [V2 L* T*] for (a); [M L? T*] for


(b) and (d); [M LI] tor (c). ‘Ine quantity on the right side of (e) has no proper dimensions since two quantities of different dimensions have been

added. Since the kinetic energy K has the dimensions of [M L? T*], formulas {a), {c) and (e)are ruled out. Note that dimensional arguments cannot tell which of the two, (b) or (d), ia the

correct formula. For this, one must turn to the actual definition of kinetic energy (see Chapter


6). The correct formula for kinetic energy is given by (b). <

2.10.2 Deducing Relation among the

Physical Quantities

The method of dimensions can sometimes be used to deduce relation among the physical quantities. For this we should know the dependence of the physical quantity on other quantities (upto three physical quantities or linearly independent variables) and consider it

as a product type of the dependence. Let us take an example.


Example 2.17 Consider a simple

pendulum, having a bob attached to a

string, that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on fis length (9, mass of the bob


(m) and acceleration due to gravity (q.

Derive the expression for ita time period using method of dimensions.

Answer The dependence of time period T on the quantities i, g and mas a product may be written as :


Take gym

where k is dimensionless constant and x, y and 2 are the exponents.


By considering dimensions on both sides, we have

[L°M°T']= IU I [’ T2 i (M! ik

=L*9T*) ME


On equating the dimensions on both sides,we have

x+y=0; -2y= 1; and z=0

x= 1 y= 1 z=0


So that «= 5 en a Then, T=k* g*

_T= Kf

on g


Note that value of constant k can not be obtained by the method of dimensions. Here it does not matter ifaome number multiplies the right side of this formula, because that does not affect its dimensions.


> fl


Actually, k = 2x so that T= anf <

Dimensional analysis is very useful in deducing Telations among the interdependent physical quantities. However, dimensionless constants

cannot be obtained by this method. The method of dimensions can only test the dimensional validity, but not the exact relationship between physical quantities in any equation. It does not distinguish between the physical quantities

having same dimensions.


A number of exercises at the end of this

chapter will help you develop skill in

dimensional analysis.



1. Phyatics ia a quantitative acience, based on measurement of physical quantities. Certain physical quantities have been chosen as fundamental or base quantities (such as length,mass, time, electric current, thermodynamic temperature, amount of substance, and

luminous intenaity).


2. Each base quaniity is defined in terms of a certain basic, arbitrarily chosen but properly standardised reference standard called unit (such as metre, kilogram, second, ampere,

kelvin, mole and candela). The units for the fundamental or base quantities are called fundamental or base units.


3. Other phyaical quantities, derived from the base quantities, can be expressed as a combination of the base unite and are called derived units. A complete set of unita,both fundamental and derived, is called a syatem of unite.


4. The International System of Unite (SI) based on seven base units is at preaent internationally accepted unit system and is widely used throughout the world.


5. The SI units are used in all physical measurements, for both the base quantities and the derived quantities obtained from them. Certain derived unite are expressed by means of SI unite with special names (such as joule, newton, watt, etc).


6. The SI units have well defined and internationally accepted unit symbols (such as m for metre, kg for kilogram, 6 for second, A for ampere, N for newton etc.).


7. Phyaical measurements are usually expressed for small and large quantities in scientific notation, with powers of 10. Scientific notation and the prefixes are used to simplify measurement notation and numerical computation, giving indication to the precision

of the numbers.


8. Certain general rules and guidelines must be followed for using notations for physical quantities and standard symbols for SI unite, some other units and SI prefixes for expreasing properly the physical quantities and measurements.


9. In computing any physical quantity, the units for derived quantities involved in the relationship(s) are treated as though they were algebraic quantities till the desired unita are obtained.


10. Direct and indirect methods can be used for the measurement of physical quantities.In measured quantities, while expressing the remult, the accuracy and precision of meaguring instruments along with errora in measurements should be taken into account.


11. In measured and computed quantities proper significant figures only should be retained.Rules for determining the number of significant figures, carrying out arithmetic operations with them, and Tounding off ' the uncertain digita must be followed.


12. The dimensions of base quantities and combination of these dimenatona describe the nature of physical quantities. Dimensional analyaia can be used to check the dimensional

consistency of equations, deducing relations among the physical quantities, etc. A dimensionally consistent equation need not be actually an exact (correct) equation,but a dimensionally wrong or inconaistent equation must be wrong.



Note : In stating numerical answers, take care of significant figures.


2.1 Fill in the Blanks

(a) The volume of a cube of side 1 cm ia equal to .....m?


(b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to -».(um)*


(c) A vehicle moving with a speed of 16 km h7 covers....m in 1 8


(d) The relative density of lead is 11.3. Ite density is ....g¢ cm or mr.


3.2 Fill in the blanks by suitable conversion of unite

(a) lkgm's* -....g cm*s*

(b) lm -..... ly

(9 3.0ms* -.... kn h*


(a) G = 6.67 x 10 N m? (kg) =... (am) sg.


3.3 Acaloric ja a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m? s*.Suppose we employ a system of units in which the unit of mass equals akg, the unit of length equals 6m, the unit of time is ya. Show that a caloric has a magnitude 4.2 a 8 y7 in tenms of the new units.


2.4 Explain this statement clearly :

“To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever mecessary :

(a) atoms are very small objects

(b) a jet plane moves with great speed

(c the mass of Jupiter is very large

(d) the air inside thie room contains a large number of molecules  a proton is much more massive than an electron

(0 the speed of sound ie much smaller than the speed of light.


2.5 Anew unit of length is chosen such that the speed of light in vacuum fa unity. What ia the distance between the Sun and the Earth in terms of the new unit if light takes 8 min and 20 s to cover this distance ?

3.6 Which of the following is the most precise device for measuring length :


(a) a vernier callipers with 20 divisione on the sliding scale


(b) a screw gauge of pitch 1 mm and 100 divisions on the circular scale


(c an optical instrument that can measure length to within a wavelength of light 7?


2.7 A student measures the thickness of a human hair by looking at it through a

microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What ta the estimate on the thickness of hair ?


2.8 Answer the following :

(a) You are given a thread and a metre scale. How will you estimate the diameter of the thread ?


(b)A acrew gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Do you think it ia possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular acale ?


(c) The mean diameter of a thin braaga rod is to be measured by vernier callipera. Why faa set of 100 measurements of the diameter expected to yield a more reliable estimate than a set of 5 measurements only ?


3.9 The photograph of a house occupies an area of 1.75 cm? on a 35 mm alide. The slide is projected on to a screen, and the area of the house on the screen is 1.55 m?, What is the linear magnification of the projector-ecreen arrangement.


2.10 State the number of significant figures in the following :

(a) 0.007 m?*

(b) 2.64 x 10% kg

(c 0.2370 g cm

(d) 6.320 J

(e) 6.032 N m*

(f) +0.0006032 m*


3.11 The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m,and 2.01 cm respectively. Give the area and volume of the sheet to correct significant



2.12 The mass of a box measured by a grocer’s balance is 2.300 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What ts (a) the total mass of the box, (b} the difference in the masses of the pieces to correct significant figures ?


2.13 A physical quantity P is related to four observable a, b, c and d as follows :P=a'b*/ (fe 4}

The percentage errors of measurement in a, Bb, c and d are 1%, 3%, 4% and 2%,

respectively. What is the percentage error in the quantity P? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result 7?


2.14 A book with many printing errors contains four dtfferent formulas for the displacement y of a particle undergoing a certain periodic motion :

(a) y= asin 2x t/T

(b) y= asin vt

(c) y = (4/7) sm t/a

(D y =(a¥2) (sin nt /T + cos 2xt /T)


(a= maximum displacement of the particle, v = speed of the particle. T= time-period of motion). Rule out the wrong formulas on dimensional grounds.


2.15 A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ m, of a particle m terms of its speed v and the speed of light, c. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation

almost correctly but forgets where to put the constant c. He writes :


meso l/2

(1-v*) ,

Guess where to put the missing c.


2.16 The unit of length convenient on the Se ee ence ee ot ae and is

denoted by A: 1 A= 107" m. The size of a hydrogen atom is about 0.5 A. What is the total atomic volume in m° of a mole of hydrogen atoms ?


2.17 One mole of an ideal gas at standard temperature and preasure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? (Take the size of hydrogen molecule to be about 1 A). Why is this ratio 80 large ?


2.18 Explain this common observation clearly : If you look out of the window of a fast moving train, the nearby trees, houses etc, seem to move rapidly in a direction oppostte to the train's motion, but the distant objects (hill tops, the Moon, the stars etc.)seam to be stationary. (In fact, since you are aware that you are moving, these

distant objects seem to move with you).


2.18 The principle of ‘parallax’ in section 2.3.1 is used in the  determination of distances of very distant stars. The baseline AB is the line joining the Earth's two locations six months apart in its orbit around the Sun. That is, the baseline ia about the diameter of the Earth's orbit «3 x 10"4m. However, even the neareat atars are so distant that with auch a long baseline, they show parallax only of the order of 1" (second) of arc or ao. A parsec is a convenient unit of length on the astronomical scale. It is the

distance of an object that will show a parallax of 1° (second) of arc from opposite ends of a baseline equal to the distance from the Earth to the Sun. How much is a parsec in terms of metres ?


2.20 The nearest star to our solar system ie 4.29 light yeare away. How much is this distance in terms of parsecs? How much parallax would this star (named Alpha Centauri) show when viewed from two locations of the Earth aix months apart in its orbit around the Sun ?


2.21 Precise measurements of physical quantities are a need of science. For example, to ascertain the speed of an aircraft, one must have an accurate method to find ite positiona at closely separated matants of time. This waa the actual motivation behind the discovery of radar in World War II. Think of different examples in modern science

where precise measurements of length, time, maas etc. are needed. Also, wherever you can, give a quantitative idea of the preciaion needed.


2.22 Just as precise measurements are necessary in science, it is equally important to be able to make rough estimates of quantities using rudimentary ideas and common observations. Think of ways by which you can catimate the following (where an

estimate is difficult to obtain, try to get an upper bound on the quantity) :


(a) the total mass of rain-bearing clouds over India during the Monsoon

(b) the massa of an elephant

(c) the wind speed during a storm

d) the number of strands of hatr on your head

(e the number of air molecules in your dassroom.


3.23 The Sun is a hot plasma Gonized mattes) with ite inner core at a taamperature exceeding 107 K, and ite outer surface at a temperature of about 6000 K. At these high temperatures, no substance remaine in a solid or liquid phase. In what range do you expect the mass density of the Sun te be, in the range of denaitics of solids and

Hquids or gases ? Check if your guess is correct from the following data : mass of the Sun = 2.0 x10” kg, radius of the Sun = 7.0 x 10° m.


2.24 When the planet Jupiter is at a distance of 824.7 million kilometers from the Earth,its angular diameter is measured to be 35.72” of arc. Calculate the diameter of Jupiter.

Additional Exorvises


2.28 Aman walking briskly in rain with speed v muet slant his umbrella forward making an angle 8 with the vertical. A student derives the following relation between @ and v: tan 6 = v and checks that the relation hae a correct limit: as v + 0, 0 30, as expected. (We are assuming there is no strong wind and that the rain falls vertically

for a stationary man). Do you think this relation can be correct ? If not, guess the correct relation.


2.26 It is claimed that two ceaium clocks, if allowed to run for 100 years, free from any disturbance, may differ by only about 0.02 s. What does this imply for the accuracy of the standard cesium clock in measuring a time-interval of 1 s ?


2.27 Estimate the average mase density of a sodium atom assuming ite size to be about


2.5 A. (Use the known values of Avogadro's number and the atomic mass of sodium).Compare it with the density of sodium in its crystalline phase : 970 kg m“. Are the two densities of the same order of magnitude ? If ao, why 7?


3.28 The unit of length convenient on the nuclear ecale is a fermi : 1 f = 10°°m. Nuclear sizes obey roughly the following anpirical relation :



where ris the radius of the nucleus, A ites mass number, and r, is a constant equal to about, 1.2 f. Show that the rule implica that nuclear mass density ia nearly constant for different nuciei. Estimate the mass denaity of sodium nucleus. Compare it with the average mass density of a sodium atom obtained in Exercise. 2.27.


2.29 A LASER is a source of very intense, monochromatic, and unidirectional beam of light. These properties of a laser light can be exploited to measure long distances.

The distance of the Moon from the Earth has been already determined very precisely ueing a laser as a eource of light. A laser ight beamed at the Moon takes 2.56 8 to return after reflection at the Moon's surface. How much is the radius of the lunar orbit around the Earth ?


2.30 A SONAR (sound navigation and ranging) uses ultrasonic waves to detect and locate objects under water. In a submarine equipped with a SONAR the time delay between generation of a probe wave and the reception of its echo after reflection from an enemy submarine is found to be 77.0 a. What ia the distance of the enemy submarine?(Speed of sound in water = 1450 m 6").


3.31 The farthest objects in our Universe discovered by modern astronomers are so distant that light emitted by them takes billions of years to reach the Earth. These objects

(cnown as quasars) have many puzzling features, which have not yet been satisfactorily explained. What is the distance in lan of a quasar from which light takes 3.0 billion years to reach us?


2.32 It is a well known fact that during a total solar eclipse the disk of the moon almost completely covers the diak of the Sun. From this fact and from the information you can gather from examples 2.3 and 2.4, determine the approximate diameter of the moon.


2.33 A great physicist of this century (P.A M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac

found that from the basic constants of atomic phyaics (c, e, maaa of electron, mass of proton) and the gravitational constant G, he could arrive at a number with the dimenaion of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe (~15 billion years). From the table of fundamental constants in this book, try to see if you too can construct thia number

(or any other intereating number you can think of ). If tte coincidence with the age of the universe were significant, what would thie imply for the constancy of fundamental constants ?