NCERT Textbook: Units & Measurements - JEE PDF Download

 Page 1


CHAPTER TWO
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 be
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 units 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 Système Internationale
d’ Unites (French for International System of Units),
abbreviated as SI. The SI, with standard scheme of symbols,
units and abbreviations, developed by the Bureau
International des Poids et measures (The International
Bureau of Weights and Measures, BIPM) in 1971 were
recently revised by the General Conference on Weights and
Measures in November 2018. The scheme is now for
2.1 Introduction
2.2 The international system of
units
2.3 Measurement of length
2.4 Measurement of mass
2.5 Measurement of time
2.6 Accuracy, precision of
instruments and errors in
measurement
2.7 Significant figures
2.8 Dimensions of physical
quantities
2.9 Dimensional formulae and
dimensional equations
2.10 Dimensional analysis and its
applications
Summary
Exercises
Additional exercises
2020-21
Page 2


CHAPTER TWO
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 be
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 units 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 Système Internationale
d’ Unites (French for International System of Units),
abbreviated as SI. The SI, with standard scheme of symbols,
units and abbreviations, developed by the Bureau
International des Poids et measures (The International
Bureau of Weights and Measures, BIPM) in 1971 were
recently revised by the General Conference on Weights and
Measures in November 2018. The scheme is now for
2.1 Introduction
2.2 The international system of
units
2.3 Measurement of length
2.4 Measurement of mass
2.5 Measurement of time
2.6 Accuracy, precision of
instruments and errors in
measurement
2.7 Significant figures
2.8 Dimensions of physical
quantities
2.9 Dimensional formulae and
dimensional equations
2.10 Dimensional analysis and its
applications
Summary
Exercises
Additional exercises
2020-21
Table 2.1   SI Base Quantities  and  Units*
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 units 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 r and (b) solid angle dO
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. 2.1(a) and
(b) respectively.  The unit for plane angle is radian with
the symbol rad and the unit for the solid angle is
steradian with the symbol sr. Both these are
dimensionless quantities.
UNITS AND MEASUREMENT 17
(a)
(b)
Fig. 2.1 Description of (a) plane angle d?
and (b) solid angle 
dO
.
Base                                                       SI Units
quantity Name Symbol Definition
Length metre m The metre, symbol m, is the SI unit of length. It is defined by taking the
fixed numerical value of the speed of light in vacuum c to be 299792458
when expressed in the unit m s
–1
, where the second is defined in terms of
the caesium frequency ??cs.
Mass kilogram kg The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the
fixed numerical value of the Planck constant h to be 6.62607015×10
–34
 when
expressed in the unit J s, which is equal to kg m
2 
s
–1
, where the metre and
the second are defined in terms of c and ??cs.
Time second s The second, symbol s, is the SI unit of time. It is defined by taking the fixed
numerical value of the caesium frequency ??cs, the unperturbed ground-
state hyperfine transition frequency of the caesium-133 atom, to be
9192631770 when expressed in the unit Hz, which is equal to s
–1
.
Electric ampere A The ampere, symbol A, is the SI unit of electric current. It is defined by
taking the fixed numerical value of the elementary charge e to be
1.602176634×10
–19
 when expressed in the unit C, which is equal to A s,
where the second is defined in terms of ??cs.
Thermo kelvin K The kelvin, symbol K, is the SI unit of thermodynamic temperature.
dynamic It is defined by taking the fixed numerical value of the Boltzmann constant
Temperature k to be 1.380649×10
–23
 when expressed in the unit J K
–1
, which is equal to
kg m
2 
s
–2 
k
–1
, where the kilogram, metre and second are defined in terms of
h, c and ??cs.
Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole
substance contains exactly 6.02214076×10
23
 elementary entities. This number is the
fixed numerical value of the Avogadro constant, N
A
, when expressed in the
unit mol
–1
 and is called the Avogadro number. The amount of substance,
symbol n, of a system is a measure of the number of specified elementary
entities. An elementary entity may be an atom, a molecule, an ion, an electron,
any other particle or specified group of particles.
Luminous candela cd The candela, symbol cd, is the SI unit of luminous intensity in given direction.
intensity It is defined by taking the fixed numerical value of the luminous efficacy of
monochromatic radiation of frequency 540×10
12
 Hz, K
cd
, to be 683 when expressed
in the unit lm W
–1
, which is equal to cd sr W
–1
, or cd sr kg
–1
m
–2
s
3
, where the
kilogram, metre and second are defined in terms of h, c and ??cs.
* The values mentioned here need not be remembered or asked in a test. They are given here only to
indicate the extent of accuracy to which they are measured. With progress in technology, the measuring
techniques get improved leading to measurements with greater precision. The definitions of base units
are revised to keep up with this progress.
2020-21
Page 3


CHAPTER TWO
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 be
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 units 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 Système Internationale
d’ Unites (French for International System of Units),
abbreviated as SI. The SI, with standard scheme of symbols,
units and abbreviations, developed by the Bureau
International des Poids et measures (The International
Bureau of Weights and Measures, BIPM) in 1971 were
recently revised by the General Conference on Weights and
Measures in November 2018. The scheme is now for
2.1 Introduction
2.2 The international system of
units
2.3 Measurement of length
2.4 Measurement of mass
2.5 Measurement of time
2.6 Accuracy, precision of
instruments and errors in
measurement
2.7 Significant figures
2.8 Dimensions of physical
quantities
2.9 Dimensional formulae and
dimensional equations
2.10 Dimensional analysis and its
applications
Summary
Exercises
Additional exercises
2020-21
Table 2.1   SI Base Quantities  and  Units*
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 units 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 r and (b) solid angle dO
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. 2.1(a) and
(b) respectively.  The unit for plane angle is radian with
the symbol rad and the unit for the solid angle is
steradian with the symbol sr. Both these are
dimensionless quantities.
UNITS AND MEASUREMENT 17
(a)
(b)
Fig. 2.1 Description of (a) plane angle d?
and (b) solid angle 
dO
.
Base                                                       SI Units
quantity Name Symbol Definition
Length metre m The metre, symbol m, is the SI unit of length. It is defined by taking the
fixed numerical value of the speed of light in vacuum c to be 299792458
when expressed in the unit m s
–1
, where the second is defined in terms of
the caesium frequency ??cs.
Mass kilogram kg The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the
fixed numerical value of the Planck constant h to be 6.62607015×10
–34
 when
expressed in the unit J s, which is equal to kg m
2 
s
–1
, where the metre and
the second are defined in terms of c and ??cs.
Time second s The second, symbol s, is the SI unit of time. It is defined by taking the fixed
numerical value of the caesium frequency ??cs, the unperturbed ground-
state hyperfine transition frequency of the caesium-133 atom, to be
9192631770 when expressed in the unit Hz, which is equal to s
–1
.
Electric ampere A The ampere, symbol A, is the SI unit of electric current. It is defined by
taking the fixed numerical value of the elementary charge e to be
1.602176634×10
–19
 when expressed in the unit C, which is equal to A s,
where the second is defined in terms of ??cs.
Thermo kelvin K The kelvin, symbol K, is the SI unit of thermodynamic temperature.
dynamic It is defined by taking the fixed numerical value of the Boltzmann constant
Temperature k to be 1.380649×10
–23
 when expressed in the unit J K
–1
, which is equal to
kg m
2 
s
–2 
k
–1
, where the kilogram, metre and second are defined in terms of
h, c and ??cs.
Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole
substance contains exactly 6.02214076×10
23
 elementary entities. This number is the
fixed numerical value of the Avogadro constant, N
A
, when expressed in the
unit mol
–1
 and is called the Avogadro number. The amount of substance,
symbol n, of a system is a measure of the number of specified elementary
entities. An elementary entity may be an atom, a molecule, an ion, an electron,
any other particle or specified group of particles.
Luminous candela cd The candela, symbol cd, is the SI unit of luminous intensity in given direction.
intensity It is defined by taking the fixed numerical value of the luminous efficacy of
monochromatic radiation of frequency 540×10
12
 Hz, K
cd
, to be 683 when expressed
in the unit lm W
–1
, which is equal to cd sr W
–1
, or cd sr kg
–1
m
–2
s
3
, where the
kilogram, metre and second are defined in terms of h, c and ??cs.
* The values mentioned here need not be remembered or asked in a test. They are given here only to
indicate the extent of accuracy to which they are measured. With progress in technology, the measuring
techniques get improved leading to measurements with greater precision. The definitions of base units
are revised to keep up with this progress.
2020-21
PHYSICS 18
Table 2.2   Some units retained for general use (Though outside SI)
Note that when mole is used, the elementary
entities  must  be specified.  These  entities
may be atoms, molecules, ions, electrons,
other particles or specified 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 units are given special names
(Appendix A  6.2 ) and some derived SI units make
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 A 6.3 for your ready
reference. 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 A8 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 example, a
metre scale is used for lengths from 10
–3 
m to 10
2
m.  A vernier callipers is used for lengths to an
accuracy of 10
–4 
m.  A screw gauge and a
spherometer can be used to measure lengths as
less as to 10
–5 
m. To measure lengths beyond these
ranges, we make use of some special indirect
methods.
2.3.1  Measurement of Large Distances
Large distances such as the distance of a planet
or a star from the earth cannot be measured
directly with a metre scale. 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 point
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 S by the parallax method, we observe it
from two different positions (observatories) A and
B on the  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 ?ASB in Fig. 2.2 represented by
symbol ? is called the parallax angle or
parallactic angle.
As the planet is very far away,  
1,
b
D
<<
 and
therefore, ? is very small.  Then we
approximately take AB as an arc of length b of a
circle with centre at S and the distance D as
2020-21
Page 4


CHAPTER TWO
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 be
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 units 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 Système Internationale
d’ Unites (French for International System of Units),
abbreviated as SI. The SI, with standard scheme of symbols,
units and abbreviations, developed by the Bureau
International des Poids et measures (The International
Bureau of Weights and Measures, BIPM) in 1971 were
recently revised by the General Conference on Weights and
Measures in November 2018. The scheme is now for
2.1 Introduction
2.2 The international system of
units
2.3 Measurement of length
2.4 Measurement of mass
2.5 Measurement of time
2.6 Accuracy, precision of
instruments and errors in
measurement
2.7 Significant figures
2.8 Dimensions of physical
quantities
2.9 Dimensional formulae and
dimensional equations
2.10 Dimensional analysis and its
applications
Summary
Exercises
Additional exercises
2020-21
Table 2.1   SI Base Quantities  and  Units*
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 units 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 r and (b) solid angle dO
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. 2.1(a) and
(b) respectively.  The unit for plane angle is radian with
the symbol rad and the unit for the solid angle is
steradian with the symbol sr. Both these are
dimensionless quantities.
UNITS AND MEASUREMENT 17
(a)
(b)
Fig. 2.1 Description of (a) plane angle d?
and (b) solid angle 
dO
.
Base                                                       SI Units
quantity Name Symbol Definition
Length metre m The metre, symbol m, is the SI unit of length. It is defined by taking the
fixed numerical value of the speed of light in vacuum c to be 299792458
when expressed in the unit m s
–1
, where the second is defined in terms of
the caesium frequency ??cs.
Mass kilogram kg The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the
fixed numerical value of the Planck constant h to be 6.62607015×10
–34
 when
expressed in the unit J s, which is equal to kg m
2 
s
–1
, where the metre and
the second are defined in terms of c and ??cs.
Time second s The second, symbol s, is the SI unit of time. It is defined by taking the fixed
numerical value of the caesium frequency ??cs, the unperturbed ground-
state hyperfine transition frequency of the caesium-133 atom, to be
9192631770 when expressed in the unit Hz, which is equal to s
–1
.
Electric ampere A The ampere, symbol A, is the SI unit of electric current. It is defined by
taking the fixed numerical value of the elementary charge e to be
1.602176634×10
–19
 when expressed in the unit C, which is equal to A s,
where the second is defined in terms of ??cs.
Thermo kelvin K The kelvin, symbol K, is the SI unit of thermodynamic temperature.
dynamic It is defined by taking the fixed numerical value of the Boltzmann constant
Temperature k to be 1.380649×10
–23
 when expressed in the unit J K
–1
, which is equal to
kg m
2 
s
–2 
k
–1
, where the kilogram, metre and second are defined in terms of
h, c and ??cs.
Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole
substance contains exactly 6.02214076×10
23
 elementary entities. This number is the
fixed numerical value of the Avogadro constant, N
A
, when expressed in the
unit mol
–1
 and is called the Avogadro number. The amount of substance,
symbol n, of a system is a measure of the number of specified elementary
entities. An elementary entity may be an atom, a molecule, an ion, an electron,
any other particle or specified group of particles.
Luminous candela cd The candela, symbol cd, is the SI unit of luminous intensity in given direction.
intensity It is defined by taking the fixed numerical value of the luminous efficacy of
monochromatic radiation of frequency 540×10
12
 Hz, K
cd
, to be 683 when expressed
in the unit lm W
–1
, which is equal to cd sr W
–1
, or cd sr kg
–1
m
–2
s
3
, where the
kilogram, metre and second are defined in terms of h, c and ??cs.
* The values mentioned here need not be remembered or asked in a test. They are given here only to
indicate the extent of accuracy to which they are measured. With progress in technology, the measuring
techniques get improved leading to measurements with greater precision. The definitions of base units
are revised to keep up with this progress.
2020-21
PHYSICS 18
Table 2.2   Some units retained for general use (Though outside SI)
Note that when mole is used, the elementary
entities  must  be specified.  These  entities
may be atoms, molecules, ions, electrons,
other particles or specified 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 units are given special names
(Appendix A  6.2 ) and some derived SI units make
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 A 6.3 for your ready
reference. 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 A8 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 example, a
metre scale is used for lengths from 10
–3 
m to 10
2
m.  A vernier callipers is used for lengths to an
accuracy of 10
–4 
m.  A screw gauge and a
spherometer can be used to measure lengths as
less as to 10
–5 
m. To measure lengths beyond these
ranges, we make use of some special indirect
methods.
2.3.1  Measurement of Large Distances
Large distances such as the distance of a planet
or a star from the earth cannot be measured
directly with a metre scale. 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 point
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 S by the parallax method, we observe it
from two different positions (observatories) A and
B on the  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 ?ASB in Fig. 2.2 represented by
symbol ? is called the parallax angle or
parallactic angle.
As the planet is very far away,  
1,
b
D
<<
 and
therefore, ? is very small.  Then we
approximately take AB as an arc of length b of a
circle with centre at S and the distance D as
2020-21
UNITS AND MEASUREMENT 19
t
t
t
t
the radius  AS = BS so that AB = b = D ?  where
?  is in radians.
   
D = 
b
?
(2.1)
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 planet
and a  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 earth.  It is the angle between
the two directions when two diametrically
opposite  points  of the planet are viewed through
the telescope.  Since D is known, the diameter d
of the planet can be determined using Eq. (2.2).
Example 2.1  Calculate the angle of
(a) 1
0
 (degree) (b) 1' (minute of arc or arcmin)
and (c) 1?(second of arc or arc second) in
radians. Use 360
0
=2p rad, 1
0
=60' and
1' = 60 ?
Answer  (a) We have 360
0 
= 2p rad
1
0
 = (p /180) rad = 1.745×10
–2
 rad
(b) 1
0
 = 60' = 1.745×10
–2
 rad
1' = 2.908×10
–4
 rad 
 2.91×10
–4
 rad
(c) 1' = 60? = 2.908×10
–4
 rad
1? = 4.847×10
–4
 rad 
 4.85×10
–6
 rad
t
Example 2.2  A man wishes to estimate
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
AO; but he finds the line of sight of C shifted
from the original line of sight by an angle ?
= 40
0 
 (? is known as ‘parallax’) estimate
the distance of the tower C from his original
position A.
Fig. 2.3
Answer  We have, parallax angle ?  = 40
0
From Fig. 2.3, AB = AC tan ?
AC = AB/tan?  = 100 m/tan 40
0
= 100 m/0.8391 = 119 m
t
Example 2.3  The moon is observed from
two diametrically opposite points A and B
on Earth.  The angle ?  subtended at the
moon by the two directions of observation
is 1
o 
54'. Given the diameter of the Earth to
be about 1.276 × × × × ×     10
7
 m, compute the
distance of the moon from the Earth.
Answer  We have ?  = 1°
 
54
'
 = 114
'
( ) ( )
-6
114 60 4.85 10 = 
' '
× × ×  rad
= 3.32 ×
-
10
2
rad,
since
  
6
1 4.85 10 . = " rad
-
×
Also  b = AB =1.276 m ×10
7
Hence from Eq. (2.1), we have the earth-moon
distance,
D b = /?
   
=
1.276 10
3.32 10
    
7
-2
×
×
8
3.84 10 m = × t
Example 2.4  The Sun’s angular diameter
is measured to be 1920''.  The distance D of
the Sun from the Earth is 1.496 × 10
11
 m.
What is the diameter of the Sun ?
Fig. 2.2   Parallax method.
2020-21
Page 5


CHAPTER TWO
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 be
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 units 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 Système Internationale
d’ Unites (French for International System of Units),
abbreviated as SI. The SI, with standard scheme of symbols,
units and abbreviations, developed by the Bureau
International des Poids et measures (The International
Bureau of Weights and Measures, BIPM) in 1971 were
recently revised by the General Conference on Weights and
Measures in November 2018. The scheme is now for
2.1 Introduction
2.2 The international system of
units
2.3 Measurement of length
2.4 Measurement of mass
2.5 Measurement of time
2.6 Accuracy, precision of
instruments and errors in
measurement
2.7 Significant figures
2.8 Dimensions of physical
quantities
2.9 Dimensional formulae and
dimensional equations
2.10 Dimensional analysis and its
applications
Summary
Exercises
Additional exercises
2020-21
Table 2.1   SI Base Quantities  and  Units*
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 units 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 r and (b) solid angle dO
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. 2.1(a) and
(b) respectively.  The unit for plane angle is radian with
the symbol rad and the unit for the solid angle is
steradian with the symbol sr. Both these are
dimensionless quantities.
UNITS AND MEASUREMENT 17
(a)
(b)
Fig. 2.1 Description of (a) plane angle d?
and (b) solid angle 
dO
.
Base                                                       SI Units
quantity Name Symbol Definition
Length metre m The metre, symbol m, is the SI unit of length. It is defined by taking the
fixed numerical value of the speed of light in vacuum c to be 299792458
when expressed in the unit m s
–1
, where the second is defined in terms of
the caesium frequency ??cs.
Mass kilogram kg The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the
fixed numerical value of the Planck constant h to be 6.62607015×10
–34
 when
expressed in the unit J s, which is equal to kg m
2 
s
–1
, where the metre and
the second are defined in terms of c and ??cs.
Time second s The second, symbol s, is the SI unit of time. It is defined by taking the fixed
numerical value of the caesium frequency ??cs, the unperturbed ground-
state hyperfine transition frequency of the caesium-133 atom, to be
9192631770 when expressed in the unit Hz, which is equal to s
–1
.
Electric ampere A The ampere, symbol A, is the SI unit of electric current. It is defined by
taking the fixed numerical value of the elementary charge e to be
1.602176634×10
–19
 when expressed in the unit C, which is equal to A s,
where the second is defined in terms of ??cs.
Thermo kelvin K The kelvin, symbol K, is the SI unit of thermodynamic temperature.
dynamic It is defined by taking the fixed numerical value of the Boltzmann constant
Temperature k to be 1.380649×10
–23
 when expressed in the unit J K
–1
, which is equal to
kg m
2 
s
–2 
k
–1
, where the kilogram, metre and second are defined in terms of
h, c and ??cs.
Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole
substance contains exactly 6.02214076×10
23
 elementary entities. This number is the
fixed numerical value of the Avogadro constant, N
A
, when expressed in the
unit mol
–1
 and is called the Avogadro number. The amount of substance,
symbol n, of a system is a measure of the number of specified elementary
entities. An elementary entity may be an atom, a molecule, an ion, an electron,
any other particle or specified group of particles.
Luminous candela cd The candela, symbol cd, is the SI unit of luminous intensity in given direction.
intensity It is defined by taking the fixed numerical value of the luminous efficacy of
monochromatic radiation of frequency 540×10
12
 Hz, K
cd
, to be 683 when expressed
in the unit lm W
–1
, which is equal to cd sr W
–1
, or cd sr kg
–1
m
–2
s
3
, where the
kilogram, metre and second are defined in terms of h, c and ??cs.
* The values mentioned here need not be remembered or asked in a test. They are given here only to
indicate the extent of accuracy to which they are measured. With progress in technology, the measuring
techniques get improved leading to measurements with greater precision. The definitions of base units
are revised to keep up with this progress.
2020-21
PHYSICS 18
Table 2.2   Some units retained for general use (Though outside SI)
Note that when mole is used, the elementary
entities  must  be specified.  These  entities
may be atoms, molecules, ions, electrons,
other particles or specified 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 units are given special names
(Appendix A  6.2 ) and some derived SI units make
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 A 6.3 for your ready
reference. 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 A8 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 example, a
metre scale is used for lengths from 10
–3 
m to 10
2
m.  A vernier callipers is used for lengths to an
accuracy of 10
–4 
m.  A screw gauge and a
spherometer can be used to measure lengths as
less as to 10
–5 
m. To measure lengths beyond these
ranges, we make use of some special indirect
methods.
2.3.1  Measurement of Large Distances
Large distances such as the distance of a planet
or a star from the earth cannot be measured
directly with a metre scale. 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 point
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 S by the parallax method, we observe it
from two different positions (observatories) A and
B on the  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 ?ASB in Fig. 2.2 represented by
symbol ? is called the parallax angle or
parallactic angle.
As the planet is very far away,  
1,
b
D
<<
 and
therefore, ? is very small.  Then we
approximately take AB as an arc of length b of a
circle with centre at S and the distance D as
2020-21
UNITS AND MEASUREMENT 19
t
t
t
t
the radius  AS = BS so that AB = b = D ?  where
?  is in radians.
   
D = 
b
?
(2.1)
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 planet
and a  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 earth.  It is the angle between
the two directions when two diametrically
opposite  points  of the planet are viewed through
the telescope.  Since D is known, the diameter d
of the planet can be determined using Eq. (2.2).
Example 2.1  Calculate the angle of
(a) 1
0
 (degree) (b) 1' (minute of arc or arcmin)
and (c) 1?(second of arc or arc second) in
radians. Use 360
0
=2p rad, 1
0
=60' and
1' = 60 ?
Answer  (a) We have 360
0 
= 2p rad
1
0
 = (p /180) rad = 1.745×10
–2
 rad
(b) 1
0
 = 60' = 1.745×10
–2
 rad
1' = 2.908×10
–4
 rad 
 2.91×10
–4
 rad
(c) 1' = 60? = 2.908×10
–4
 rad
1? = 4.847×10
–4
 rad 
 4.85×10
–6
 rad
t
Example 2.2  A man wishes to estimate
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
AO; but he finds the line of sight of C shifted
from the original line of sight by an angle ?
= 40
0 
 (? is known as ‘parallax’) estimate
the distance of the tower C from his original
position A.
Fig. 2.3
Answer  We have, parallax angle ?  = 40
0
From Fig. 2.3, AB = AC tan ?
AC = AB/tan?  = 100 m/tan 40
0
= 100 m/0.8391 = 119 m
t
Example 2.3  The moon is observed from
two diametrically opposite points A and B
on Earth.  The angle ?  subtended at the
moon by the two directions of observation
is 1
o 
54'. Given the diameter of the Earth to
be about 1.276 × × × × ×     10
7
 m, compute the
distance of the moon from the Earth.
Answer  We have ?  = 1°
 
54
'
 = 114
'
( ) ( )
-6
114 60 4.85 10 = 
' '
× × ×  rad
= 3.32 ×
-
10
2
rad,
since
  
6
1 4.85 10 . = " rad
-
×
Also  b = AB =1.276 m ×10
7
Hence from Eq. (2.1), we have the earth-moon
distance,
D b = /?
   
=
1.276 10
3.32 10
    
7
-2
×
×
8
3.84 10 m = × t
Example 2.4  The Sun’s angular diameter
is measured to be 1920''.  The distance D of
the Sun from the Earth is 1.496 × 10
11
 m.
What is the diameter of the Sun ?
Fig. 2.2   Parallax method.
2020-21
PHYSICS 20
t
Answer   Sun’s angular diameter a
   = 1920"
   
= × ×
-
1920 4.85 10 rad
6
   
= ×
-
9.31 10 rad
3
Sun’s diameter
                     
d D = a 
                        
3 11
9.31 10 1.496 10 m
? ? ? ?
? ? ? ?
? ? ? ?
-
= × × ×
     
9
=1.39 10 m ×
t
2.3.2 Estimation of Very Small Distances:
Size of a Molecule
To measure a very small size, like that of a
molecule (10
–8
 m to 10
–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 Class XII
Physics textbook). For visible light the range of
wavelengths  is from about 4000 Å to 7000 Å
(1 angstrom = 1 Å = 10
-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 Å have been
built. They can almost resolve atoms and
molecules in a material. In recent times,
tunnelling microscopy has been developed in
which again the limit of resolution is better than
an angstrom. It is possible to estimate the sizes
of molecules.
A simple 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
–9 
m.
The idea is to first form mono-molecular layer
of oleic acid on water surface.
We dissolve 1 cm
3
 of oleic acid in alcohol to
make a solution of 20 cm
3
. Then we take 1 cm
3
of this solution and dilute it to 20 cm
3
, using
alcohol. So, the concentration of the solution is
equal to 
1
20 20
3
×
?
?
?
?
?
?
cm
 
of oleic acid/cm
3
 of
solution. Next we lightly sprinkle some
lycopodium powder on the surface of water in a
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 (V cm
3
).
Volume of n drops of solution
                             = nV cm
3
Amount of oleic acid in this solution
                             = 
nV 
1
20 20 ×
?
?
?
?
?
? cm
3
This solution of oleic acid spreads very fast
on the surface of water and forms a very thin
layer of thickness t.  If this spreads to form a
film of area A cm
2
, then the thickness of the
film
t =
Volume of the film
Area of the film
or,
cm
20 20 
nV
t
A
=
×
(2.3)
If we assume that the film has mono-molecular
thickness, then this becomes the size or diameter
of a molecule of oleic acid. The value of this
thickness comes out to be of the order of 10
–9
 m.
Example 2.5  If the size of a nucleus (in
the range of 10
–15
 to 10
–14 
m) 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
–5
m to 10
–4
m.
Answer  The size of a nucleus is in the range of
10
–15 
m and 10
–14
 m.  The tip of a sharp pin is
taken to be in the range of 10
–5
 m and 10
–4
 m.
Thus we are scaling up by a factor of 10
10
. An
atom roughly of size 10
–10
 m will be scaled up to a
size of 1 m. Thus a nucleus in an atom is as small
in size as the tip of a sharp pin placed at the centre
of a sphere of radius about a metre long.
t
2020-21