# Chapter 2 Units And Measurement

**CHAPTER NO.2 UNITS AND MEASUREMENT **

**2.1 INTRODUCTION**

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.

**2.2 THE INTERNATIONAL 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.

**2.3 MEASUREMENT OF LENGTH**

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

methods.

**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

d=aD

=(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)

**2.4 MEASUREMENT OF MASS**

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.

**2.6 MEASUREMENT OF TIME**

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 ?

**2.6 ACCURACY, PRECISION OF INSTRUMENTS**

**AND ERRORS IN MEASUREMENT**

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

quantity.

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...

Thus,

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

=0.1ls

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

T=2.6+0.1l8

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

2.6.2 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

Z2AZ=(A#AA) (B+AB = AB+ BAA+ AAB+AAAB.

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.)

4

(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% <

2.7 SIGNIFICANT FIGURES

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

significant.

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

measurement.

(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

Places.

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 %

=£0.1%

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.

**2.8 DIMENSIONS OF PHYSICAL QUANTITIES**

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*].

**2.0 DIMENSIONAL FORMULAE AND**

**DIMENSIONAL EQUATIONS**

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

Teference.

**2.10 DIMENSIONAL ANALYSIS AND ITS**

**APPLICATIONS**

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

comrect.

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.

**SUMMARY**

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.

**EXERCISES**

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 ....kg 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

figures.

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 :

Mg

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 :

rerAVv

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 ?

0comments