NCERT Textbook: Units & Measurement | Physics Class 11 - NEET PDF Download

 Page 1


CHAPTER ONE
UNITS AND MEASUREMENT
1.1  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.
1.2  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
1.1 Introduction
1.2 The international system of
units
1.3 Significant figures
1.4 Dimensions of physical
quantities
1.5 Dimensional formulae and
dimensional equations
1.6 Dimensional analysis and its
applications
Summary
Exercises
2024-25
Page 2


CHAPTER ONE
UNITS AND MEASUREMENT
1.1  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.
1.2  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
1.1 Introduction
1.2 The international system of
units
1.3 Significant figures
1.4 Dimensions of physical
quantities
1.5 Dimensional formulae and
dimensional equations
1.6 Dimensional analysis and its
applications
Summary
Exercises
2024-25
PHYSICS 2
Table 1.1   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 1.1. 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. 1.1(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.
(a)
(b)
Fig. 1.1 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.
2024-25
Page 3


CHAPTER ONE
UNITS AND MEASUREMENT
1.1  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.
1.2  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
1.1 Introduction
1.2 The international system of
units
1.3 Significant figures
1.4 Dimensions of physical
quantities
1.5 Dimensional formulae and
dimensional equations
1.6 Dimensional analysis and its
applications
Summary
Exercises
2024-25
PHYSICS 2
Table 1.1   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 1.1. 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. 1.1(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.
(a)
(b)
Fig. 1.1 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.
2024-25
UNITS AND MEASUREMENT 3
Table 1.2   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 1.2.
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.
1.3 SIGNIFICANT FIGURES
As discussed above, every measurement
involves errors. 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 number that includes all digits 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
reported after measurement to be 287.5 cm has
four significant figures, the digits 2, 8, 7 are
certain while the digit 5 is uncertain. Clearly,
reporting the result of measurement that
includes more digits than the significant digits
is superfluous 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 important
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
mm or 23080 µm.
All these numbers have the same number of
significant figures (digits 2, 3, 0, 8), namely four.
2024-25
Page 4


CHAPTER ONE
UNITS AND MEASUREMENT
1.1  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.
1.2  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
1.1 Introduction
1.2 The international system of
units
1.3 Significant figures
1.4 Dimensions of physical
quantities
1.5 Dimensional formulae and
dimensional equations
1.6 Dimensional analysis and its
applications
Summary
Exercises
2024-25
PHYSICS 2
Table 1.1   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 1.1. 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. 1.1(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.
(a)
(b)
Fig. 1.1 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.
2024-25
UNITS AND MEASUREMENT 3
Table 1.2   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 1.2.
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.
1.3 SIGNIFICANT FIGURES
As discussed above, every measurement
involves errors. 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 number that includes all digits 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
reported after measurement to be 287.5 cm has
four significant figures, the digits 2, 8, 7 are
certain while the digit 5 is uncertain. Clearly,
reporting the result of measurement that
includes more digits than the significant digits
is superfluous 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 important
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
mm or 23080 µm.
All these numbers have the same number of
significant figures (digits 2, 3, 0, 8), namely four.
2024-25
PHYSICS 4
This shows that the location of decimal 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-zero digits
are significant, no matter where the
decimal point is, if at all.
• If the number is less than 1, the zero(s)
on the right of decimal point but to the
left of the first non-zero digit are not
significant. [In 0.00 2308, the underlined
zeroes are not significant].
• The terminal or trailing zero(s) in a
number without a decimal point are not
significant.
[Thus 123 m = 12300 cm = 123000 mm has
three significant figures, the trailing zero(s)
being not significant.] However, you can also
see the next observation.
• The trailing zero(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, it 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 cm = 4700 mm = 0.004700 km
Since the last number has trailing 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 × 10
b
, 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
=
10). Then the number can be
expressed approximately as 10
b
 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
b
. For example, the diameter of the
earth (1.28×10
7
m) is of the order of 10
7
m with
the order of magnitude 7. The diameter of
hydrogen atom (1.06 ×10
–10
m) is of the order of
10
–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 decimal after
the first digit. Now the confusion mentioned in
(a) above disappears :
 4.700 m = 4.700 × 10
2
 cm
        = 4.700 × 10
3
 mm = 4.700 × 10
–3
 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 confusion
arises about the trailing zero(s) in the base
number 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 :
• For a number greater than 1, without any
decimal, the trailing zero(s) are not
significant.
• For a number with a decimal, the trailing
zero(s) are significant.
(5) The digit 0 conventionally put on the left of a
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
representing measured values are exact and
have infinite number of significant digits. For
example in 
2
d
r =
 or  s = 2pr, the factor 2 is an
exact number and it can be written as 2.0, 2.00
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Page 5


CHAPTER ONE
UNITS AND MEASUREMENT
1.1  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.
1.2  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
1.1 Introduction
1.2 The international system of
units
1.3 Significant figures
1.4 Dimensions of physical
quantities
1.5 Dimensional formulae and
dimensional equations
1.6 Dimensional analysis and its
applications
Summary
Exercises
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PHYSICS 2
Table 1.1   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 1.1. 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. 1.1(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.
(a)
(b)
Fig. 1.1 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.
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UNITS AND MEASUREMENT 3
Table 1.2   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 1.2.
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.
1.3 SIGNIFICANT FIGURES
As discussed above, every measurement
involves errors. 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 number that includes all digits 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
reported after measurement to be 287.5 cm has
four significant figures, the digits 2, 8, 7 are
certain while the digit 5 is uncertain. Clearly,
reporting the result of measurement that
includes more digits than the significant digits
is superfluous 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 important
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
mm or 23080 µm.
All these numbers have the same number of
significant figures (digits 2, 3, 0, 8), namely four.
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PHYSICS 4
This shows that the location of decimal 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-zero digits
are significant, no matter where the
decimal point is, if at all.
• If the number is less than 1, the zero(s)
on the right of decimal point but to the
left of the first non-zero digit are not
significant. [In 0.00 2308, the underlined
zeroes are not significant].
• The terminal or trailing zero(s) in a
number without a decimal point are not
significant.
[Thus 123 m = 12300 cm = 123000 mm has
three significant figures, the trailing zero(s)
being not significant.] However, you can also
see the next observation.
• The trailing zero(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, it 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 cm = 4700 mm = 0.004700 km
Since the last number has trailing 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 × 10
b
, 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
=
10). Then the number can be
expressed approximately as 10
b
 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
b
. For example, the diameter of the
earth (1.28×10
7
m) is of the order of 10
7
m with
the order of magnitude 7. The diameter of
hydrogen atom (1.06 ×10
–10
m) is of the order of
10
–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 decimal after
the first digit. Now the confusion mentioned in
(a) above disappears :
 4.700 m = 4.700 × 10
2
 cm
        = 4.700 × 10
3
 mm = 4.700 × 10
–3
 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 confusion
arises about the trailing zero(s) in the base
number 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 :
• For a number greater than 1, without any
decimal, the trailing zero(s) are not
significant.
• For a number with a decimal, the trailing
zero(s) are significant.
(5) The digit 0 conventionally put on the left of a
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
representing measured values are exact and
have infinite number of significant digits. For
example in 
2
d
r =
 or  s = 2pr, the factor 2 is an
exact number and it can be written as 2.0, 2.00
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UNITS AND MEASUREMENT 5
or 2.0000 as required. Similarly, in 
t
T
n
=
, n is
an exact number.
1.3.1 Rules for Arithmetic Operations with
Significant Figures
The result of a calculation involving approximate
measured values of quantities (i.e. values with
limited 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, if mass of an object is measured
to be, say, 4.237 g (four significant figures) and
its volume is measured to be 2.51 cm
3
, then its
density, by mere arithmetic division, is
1.68804780876 g/cm
3
 upto 11 decimal places.
It would be clearly absurd and irrelevant to
record the calculated value of density to such a
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 significant
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.
Density
4.237g
2.51 cm
1.69 g cm
3
-3
= =
Similarly,  if the speed of light is given as
3.00 × 10
8
 m s
-1
 (three significant figure) and
one year (1y = 365.25 d) has 3.1557 × 10
7
 s (five
significant figures), the light year is 9.47 × 10
15
 m
(three significant figures).
(2) In addition or subtraction, the final result
should 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 g, 227.2 g and 0.301 g by mere arithmetic
addition,  is 663.821 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.
Similarly, the difference in length can be
expressed as :
0.307 m – 0.304 m = 0.003 m = 3 ×10
–3
 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 × 10
–3
 m in the example of subtraction. They
do not convey the precision of measurement
properly. For addition and subtraction, the rule
is in terms of decimal places.
1.3.2   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 1.75, while the number 1.743 would
be 1.74.  The rule by convention is that the
preceding digit is raised by 1 if the
insignificant digit to be dropped (the
underlined digit in this case)  is more than
5, and is left  unchanged if the latter is less
than 5.  But what if the number is 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 number 2.745 rounded
off to three significant figures becomes 1.74.  On
the other hand, the number 2.735 rounded off
to three significant figures becomes 1.74 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
known to be within many significant figures,
such as in 1.99792458  × 10
8
 m/s for the speed
of light in vacuum, is rounded off to an
approximate value 3 × 10
8
 m/s , which is often
employed in computations.  Finally, remember
that exact numbers that appear in formulae like
2 p in T
L
g
= 2p , have a large (infinite) number
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