Units and Dimensions | Physics Optional Notes for UPSC PDF Download

 Page 1


UNIT AND DIMENSION 
Physics is about studying nature and the laws that govern it. Richard Feynman provided 
a great analogy for understanding nature. Imagine not knowing the rules of chess but 
being able to observe players' moves. Over time, you might figure out some rules. With 
these rules, you could analyze why a player made a specific move, although it might be 
challenging. Even with full knowledge of chess rules, it's not easy to grasp all the 
complexities in a game and predict the best move in every situation. Nonetheless, 
understanding the fundamental rules is essential to make any progress. 
Similarly, when observing a chess game without knowing all the rules, one might 
incorrectly guess some rules based on partial information. An experienced player may 
then make a move that surprises the observer, revealing new rules or disproving old 
ones. 
Physics operates similarly. The natural world is like a vast chess game played by Nature, 
and the events within it are akin to moves in this grand game. We observe these events 
and try to deduce the basic rules governing them. Sometimes, we encounter new events 
that don't align with our previous rules, forcing us to reconsider and revise our 
understanding, leading to the discovery of new rules. 
UNITS AND DIMENSIONS 
Physical quantities: Ail quantities that can be measured are called physical quantities. 
E.g. time, length, mass, force, work done, etc. In physics, we study physical quantities 
and their interrelationships. 
Measurement: Measurement is the comparison of a quantity with a standard of the 
same physical quantity. 
Different countries & states followed different standards. This created confusion. 
It was then when some extraordinary people created a committee called SI (Systeme 
International) they defined the standards of physical quantities and called them the 
UNITS. 
Unit: All physical quantities are measured w.r.t. the standard magnitude of the same 
physical quantity and these standards are called UNITS. E.g. second, meter, kilogram, 
etc. 
So the four basic properties of units are:- 
1. They must be well-defined. 
2. They should be easily available and reproducible. 
3. They should be invariable e.g. step as a unit of length is not invariable. 
4. They should be accepted to all. 
Page 2


UNIT AND DIMENSION 
Physics is about studying nature and the laws that govern it. Richard Feynman provided 
a great analogy for understanding nature. Imagine not knowing the rules of chess but 
being able to observe players' moves. Over time, you might figure out some rules. With 
these rules, you could analyze why a player made a specific move, although it might be 
challenging. Even with full knowledge of chess rules, it's not easy to grasp all the 
complexities in a game and predict the best move in every situation. Nonetheless, 
understanding the fundamental rules is essential to make any progress. 
Similarly, when observing a chess game without knowing all the rules, one might 
incorrectly guess some rules based on partial information. An experienced player may 
then make a move that surprises the observer, revealing new rules or disproving old 
ones. 
Physics operates similarly. The natural world is like a vast chess game played by Nature, 
and the events within it are akin to moves in this grand game. We observe these events 
and try to deduce the basic rules governing them. Sometimes, we encounter new events 
that don't align with our previous rules, forcing us to reconsider and revise our 
understanding, leading to the discovery of new rules. 
UNITS AND DIMENSIONS 
Physical quantities: Ail quantities that can be measured are called physical quantities. 
E.g. time, length, mass, force, work done, etc. In physics, we study physical quantities 
and their interrelationships. 
Measurement: Measurement is the comparison of a quantity with a standard of the 
same physical quantity. 
Different countries & states followed different standards. This created confusion. 
It was then when some extraordinary people created a committee called SI (Systeme 
International) they defined the standards of physical quantities and called them the 
UNITS. 
Unit: All physical quantities are measured w.r.t. the standard magnitude of the same 
physical quantity and these standards are called UNITS. E.g. second, meter, kilogram, 
etc. 
So the four basic properties of units are:- 
1. They must be well-defined. 
2. They should be easily available and reproducible. 
3. They should be invariable e.g. step as a unit of length is not invariable. 
4. They should be accepted to all. 
FPS or British Engineering system : 
In this system length, mass, and time are taken as fundamental quantities and their base 
units are foot (ft), pound (lb) and second (s) respectively. 
2. CGS or Gaussian system : 
In this system, the fundamental quantities are length, mass, and time and their 
respective units are centimeters ( cm ) , grams ( g) , and second (s). 
3. MKS system : 
In this system also the fundamental quantities are length, mass, and time but their 
fundamental units are meter ( m) , kilogram ( kg ), and second (s) respectively. 
4. International system (SI) of units : 
 
This system is a modification of the MKS system and is also referred to as the Rationalized 
MKS system. It includes the three base units of the MKS system along with four 
fundamental units and two supplementary units. 
CLASSIFICATION OF UNITS 
The units of physical quantities can be classified as follows: 
1. Fundamental or base units: 
The units of fundamental quantities are called base units. In SI there are seven base 
units. 
SI BASE QUANTITIES AND THEIR UNITS 
S.No. Physical quantity SI unit Symbol 
1. Length meter m 
2. Mass kilogram kg 
3. Time second s 
4. Temperature Kelvin K 
5. Electric current ampere A 
6. Luminous intensity candela cd 
Page 3


UNIT AND DIMENSION 
Physics is about studying nature and the laws that govern it. Richard Feynman provided 
a great analogy for understanding nature. Imagine not knowing the rules of chess but 
being able to observe players' moves. Over time, you might figure out some rules. With 
these rules, you could analyze why a player made a specific move, although it might be 
challenging. Even with full knowledge of chess rules, it's not easy to grasp all the 
complexities in a game and predict the best move in every situation. Nonetheless, 
understanding the fundamental rules is essential to make any progress. 
Similarly, when observing a chess game without knowing all the rules, one might 
incorrectly guess some rules based on partial information. An experienced player may 
then make a move that surprises the observer, revealing new rules or disproving old 
ones. 
Physics operates similarly. The natural world is like a vast chess game played by Nature, 
and the events within it are akin to moves in this grand game. We observe these events 
and try to deduce the basic rules governing them. Sometimes, we encounter new events 
that don't align with our previous rules, forcing us to reconsider and revise our 
understanding, leading to the discovery of new rules. 
UNITS AND DIMENSIONS 
Physical quantities: Ail quantities that can be measured are called physical quantities. 
E.g. time, length, mass, force, work done, etc. In physics, we study physical quantities 
and their interrelationships. 
Measurement: Measurement is the comparison of a quantity with a standard of the 
same physical quantity. 
Different countries & states followed different standards. This created confusion. 
It was then when some extraordinary people created a committee called SI (Systeme 
International) they defined the standards of physical quantities and called them the 
UNITS. 
Unit: All physical quantities are measured w.r.t. the standard magnitude of the same 
physical quantity and these standards are called UNITS. E.g. second, meter, kilogram, 
etc. 
So the four basic properties of units are:- 
1. They must be well-defined. 
2. They should be easily available and reproducible. 
3. They should be invariable e.g. step as a unit of length is not invariable. 
4. They should be accepted to all. 
FPS or British Engineering system : 
In this system length, mass, and time are taken as fundamental quantities and their base 
units are foot (ft), pound (lb) and second (s) respectively. 
2. CGS or Gaussian system : 
In this system, the fundamental quantities are length, mass, and time and their 
respective units are centimeters ( cm ) , grams ( g) , and second (s). 
3. MKS system : 
In this system also the fundamental quantities are length, mass, and time but their 
fundamental units are meter ( m) , kilogram ( kg ), and second (s) respectively. 
4. International system (SI) of units : 
 
This system is a modification of the MKS system and is also referred to as the Rationalized 
MKS system. It includes the three base units of the MKS system along with four 
fundamental units and two supplementary units. 
CLASSIFICATION OF UNITS 
The units of physical quantities can be classified as follows: 
1. Fundamental or base units: 
The units of fundamental quantities are called base units. In SI there are seven base 
units. 
SI BASE QUANTITIES AND THEIR UNITS 
S.No. Physical quantity SI unit Symbol 
1. Length meter m 
2. Mass kilogram kg 
3. Time second s 
4. Temperature Kelvin K 
5. Electric current ampere A 
6. Luminous intensity candela cd 
7. Amount of substance mole mol 
 
2. Derived units: 
The units of derived quantities or the units that can be expressed in terms of the base 
units are called derived units. 
Ex. Unit of speed =
 unit of distance 
 unit of time 
=
 meter 
 second 
= ms
-1
 
Some derived units are named in honor of great scientists. 
? Unit of force - newton ( N) - Unit of frequency - hertz (Hz) etc. 
 eg. ?  Area = l × b = m
2
 Volume = l × b × h = m
3
 Density =
?? ?? =
????
?? 3
= ???? /?? 3
= ???? - ?? -3
. [?? 1
?? -3
?? 0
]
 Force = ?? × ?? = ???? ×
?? sec
2
= ???? - ?? - ?? -2
 [?? 1
?? ?? ?? -2
]
 Pressure =
 Force 
 Area 
=
kg- m - sec
2
 m
2
= kg- m
-1
- sec
-2
[M
1
 L
-1
 T
-2
]
 
DIMENSIONS 
Dimensions of a physical quantity are the powers or exponents used to express that 
quantity in terms of base quantities. For example, let's take the physical quantity force. 
Force can be defined as the product of mass and acceleration. Acceleration is the change 
in velocity per unit time, and velocity is the displacement per unit time. Therefore, the 
dimensions of force can be derived as follows: 
force = mass × acceleration 
= mass ×
 velocity 
 time 
= mass ×
 length / time 
 time 
= mass × length × ( time )
-2
 
Thus, the dimensions of force are 1 in mass, 1 in length and -2 in time. The dimensions in 
all other base quantities are zero. 
1. Dimensional formula : 
The physical quantity that is expressed in terms of the base quantities is enclosed in 
square brackets to remind us that the equation is among the dimensions and not among 
the magnitudes. Thus above equation may be written as [force] = MLT  
-2
. 
Page 4


UNIT AND DIMENSION 
Physics is about studying nature and the laws that govern it. Richard Feynman provided 
a great analogy for understanding nature. Imagine not knowing the rules of chess but 
being able to observe players' moves. Over time, you might figure out some rules. With 
these rules, you could analyze why a player made a specific move, although it might be 
challenging. Even with full knowledge of chess rules, it's not easy to grasp all the 
complexities in a game and predict the best move in every situation. Nonetheless, 
understanding the fundamental rules is essential to make any progress. 
Similarly, when observing a chess game without knowing all the rules, one might 
incorrectly guess some rules based on partial information. An experienced player may 
then make a move that surprises the observer, revealing new rules or disproving old 
ones. 
Physics operates similarly. The natural world is like a vast chess game played by Nature, 
and the events within it are akin to moves in this grand game. We observe these events 
and try to deduce the basic rules governing them. Sometimes, we encounter new events 
that don't align with our previous rules, forcing us to reconsider and revise our 
understanding, leading to the discovery of new rules. 
UNITS AND DIMENSIONS 
Physical quantities: Ail quantities that can be measured are called physical quantities. 
E.g. time, length, mass, force, work done, etc. In physics, we study physical quantities 
and their interrelationships. 
Measurement: Measurement is the comparison of a quantity with a standard of the 
same physical quantity. 
Different countries & states followed different standards. This created confusion. 
It was then when some extraordinary people created a committee called SI (Systeme 
International) they defined the standards of physical quantities and called them the 
UNITS. 
Unit: All physical quantities are measured w.r.t. the standard magnitude of the same 
physical quantity and these standards are called UNITS. E.g. second, meter, kilogram, 
etc. 
So the four basic properties of units are:- 
1. They must be well-defined. 
2. They should be easily available and reproducible. 
3. They should be invariable e.g. step as a unit of length is not invariable. 
4. They should be accepted to all. 
FPS or British Engineering system : 
In this system length, mass, and time are taken as fundamental quantities and their base 
units are foot (ft), pound (lb) and second (s) respectively. 
2. CGS or Gaussian system : 
In this system, the fundamental quantities are length, mass, and time and their 
respective units are centimeters ( cm ) , grams ( g) , and second (s). 
3. MKS system : 
In this system also the fundamental quantities are length, mass, and time but their 
fundamental units are meter ( m) , kilogram ( kg ), and second (s) respectively. 
4. International system (SI) of units : 
 
This system is a modification of the MKS system and is also referred to as the Rationalized 
MKS system. It includes the three base units of the MKS system along with four 
fundamental units and two supplementary units. 
CLASSIFICATION OF UNITS 
The units of physical quantities can be classified as follows: 
1. Fundamental or base units: 
The units of fundamental quantities are called base units. In SI there are seven base 
units. 
SI BASE QUANTITIES AND THEIR UNITS 
S.No. Physical quantity SI unit Symbol 
1. Length meter m 
2. Mass kilogram kg 
3. Time second s 
4. Temperature Kelvin K 
5. Electric current ampere A 
6. Luminous intensity candela cd 
7. Amount of substance mole mol 
 
2. Derived units: 
The units of derived quantities or the units that can be expressed in terms of the base 
units are called derived units. 
Ex. Unit of speed =
 unit of distance 
 unit of time 
=
 meter 
 second 
= ms
-1
 
Some derived units are named in honor of great scientists. 
? Unit of force - newton ( N) - Unit of frequency - hertz (Hz) etc. 
 eg. ?  Area = l × b = m
2
 Volume = l × b × h = m
3
 Density =
?? ?? =
????
?? 3
= ???? /?? 3
= ???? - ?? -3
. [?? 1
?? -3
?? 0
]
 Force = ?? × ?? = ???? ×
?? sec
2
= ???? - ?? - ?? -2
 [?? 1
?? ?? ?? -2
]
 Pressure =
 Force 
 Area 
=
kg- m - sec
2
 m
2
= kg- m
-1
- sec
-2
[M
1
 L
-1
 T
-2
]
 
DIMENSIONS 
Dimensions of a physical quantity are the powers or exponents used to express that 
quantity in terms of base quantities. For example, let's take the physical quantity force. 
Force can be defined as the product of mass and acceleration. Acceleration is the change 
in velocity per unit time, and velocity is the displacement per unit time. Therefore, the 
dimensions of force can be derived as follows: 
force = mass × acceleration 
= mass ×
 velocity 
 time 
= mass ×
 length / time 
 time 
= mass × length × ( time )
-2
 
Thus, the dimensions of force are 1 in mass, 1 in length and -2 in time. The dimensions in 
all other base quantities are zero. 
1. Dimensional formula : 
The physical quantity that is expressed in terms of the base quantities is enclosed in 
square brackets to remind us that the equation is among the dimensions and not among 
the magnitudes. Thus above equation may be written as [force] = MLT  
-2
. 
Such an expression for a physical quantity in terms of the base quantities is called the 
dimensional force, This, the dimensional formula of force is MLT  
-2
. The two versions 
given below are equivalent and are 1 interchangeably. 
(a) The dimensional formula of force is MLT
-2
. 
(b) The dimensions of force are 1 in mass, 1 in length and -2 in time. 
Thus dimensional formula of any physical quantity is that expression that represents 
how and which of the base quantities are included in that quantity. 
Mass ? M 
b. Length ? L 
c.   Time ? T 
Temp. ? ?? , K 
Current ? I, A 
f. Amount of substance ? Mol 
Linear Momentum p = M × V 
? kg· m - sec
-1
 
= [M
1
 L
1
 T
-1
] 
Kinetic energy K =
1
2
MV
2
 
= M
1
× M
0
 L
1
 T
-1
× M
0
 L
1
 T
-1
 
= ?? 1
?? 2
?? -2
 
 MKS CGS 
_. Newton 10
5
 Dyne 
 
Question. Power ( P) =
 Energy 
 Time 
=
 Work 
 Time 
=
kg-m
2
-sec
-2
sec
 
 =
?? 1
?? 2
?? -2
?? 1
= ???? - ?? 2
- ?? -3
 = [?? 1
?? 2
?? -3
]
 
Question :Torque ?? ( Tau ) = Force × Distance 
Page 5


UNIT AND DIMENSION 
Physics is about studying nature and the laws that govern it. Richard Feynman provided 
a great analogy for understanding nature. Imagine not knowing the rules of chess but 
being able to observe players' moves. Over time, you might figure out some rules. With 
these rules, you could analyze why a player made a specific move, although it might be 
challenging. Even with full knowledge of chess rules, it's not easy to grasp all the 
complexities in a game and predict the best move in every situation. Nonetheless, 
understanding the fundamental rules is essential to make any progress. 
Similarly, when observing a chess game without knowing all the rules, one might 
incorrectly guess some rules based on partial information. An experienced player may 
then make a move that surprises the observer, revealing new rules or disproving old 
ones. 
Physics operates similarly. The natural world is like a vast chess game played by Nature, 
and the events within it are akin to moves in this grand game. We observe these events 
and try to deduce the basic rules governing them. Sometimes, we encounter new events 
that don't align with our previous rules, forcing us to reconsider and revise our 
understanding, leading to the discovery of new rules. 
UNITS AND DIMENSIONS 
Physical quantities: Ail quantities that can be measured are called physical quantities. 
E.g. time, length, mass, force, work done, etc. In physics, we study physical quantities 
and their interrelationships. 
Measurement: Measurement is the comparison of a quantity with a standard of the 
same physical quantity. 
Different countries & states followed different standards. This created confusion. 
It was then when some extraordinary people created a committee called SI (Systeme 
International) they defined the standards of physical quantities and called them the 
UNITS. 
Unit: All physical quantities are measured w.r.t. the standard magnitude of the same 
physical quantity and these standards are called UNITS. E.g. second, meter, kilogram, 
etc. 
So the four basic properties of units are:- 
1. They must be well-defined. 
2. They should be easily available and reproducible. 
3. They should be invariable e.g. step as a unit of length is not invariable. 
4. They should be accepted to all. 
FPS or British Engineering system : 
In this system length, mass, and time are taken as fundamental quantities and their base 
units are foot (ft), pound (lb) and second (s) respectively. 
2. CGS or Gaussian system : 
In this system, the fundamental quantities are length, mass, and time and their 
respective units are centimeters ( cm ) , grams ( g) , and second (s). 
3. MKS system : 
In this system also the fundamental quantities are length, mass, and time but their 
fundamental units are meter ( m) , kilogram ( kg ), and second (s) respectively. 
4. International system (SI) of units : 
 
This system is a modification of the MKS system and is also referred to as the Rationalized 
MKS system. It includes the three base units of the MKS system along with four 
fundamental units and two supplementary units. 
CLASSIFICATION OF UNITS 
The units of physical quantities can be classified as follows: 
1. Fundamental or base units: 
The units of fundamental quantities are called base units. In SI there are seven base 
units. 
SI BASE QUANTITIES AND THEIR UNITS 
S.No. Physical quantity SI unit Symbol 
1. Length meter m 
2. Mass kilogram kg 
3. Time second s 
4. Temperature Kelvin K 
5. Electric current ampere A 
6. Luminous intensity candela cd 
7. Amount of substance mole mol 
 
2. Derived units: 
The units of derived quantities or the units that can be expressed in terms of the base 
units are called derived units. 
Ex. Unit of speed =
 unit of distance 
 unit of time 
=
 meter 
 second 
= ms
-1
 
Some derived units are named in honor of great scientists. 
? Unit of force - newton ( N) - Unit of frequency - hertz (Hz) etc. 
 eg. ?  Area = l × b = m
2
 Volume = l × b × h = m
3
 Density =
?? ?? =
????
?? 3
= ???? /?? 3
= ???? - ?? -3
. [?? 1
?? -3
?? 0
]
 Force = ?? × ?? = ???? ×
?? sec
2
= ???? - ?? - ?? -2
 [?? 1
?? ?? ?? -2
]
 Pressure =
 Force 
 Area 
=
kg- m - sec
2
 m
2
= kg- m
-1
- sec
-2
[M
1
 L
-1
 T
-2
]
 
DIMENSIONS 
Dimensions of a physical quantity are the powers or exponents used to express that 
quantity in terms of base quantities. For example, let's take the physical quantity force. 
Force can be defined as the product of mass and acceleration. Acceleration is the change 
in velocity per unit time, and velocity is the displacement per unit time. Therefore, the 
dimensions of force can be derived as follows: 
force = mass × acceleration 
= mass ×
 velocity 
 time 
= mass ×
 length / time 
 time 
= mass × length × ( time )
-2
 
Thus, the dimensions of force are 1 in mass, 1 in length and -2 in time. The dimensions in 
all other base quantities are zero. 
1. Dimensional formula : 
The physical quantity that is expressed in terms of the base quantities is enclosed in 
square brackets to remind us that the equation is among the dimensions and not among 
the magnitudes. Thus above equation may be written as [force] = MLT  
-2
. 
Such an expression for a physical quantity in terms of the base quantities is called the 
dimensional force, This, the dimensional formula of force is MLT  
-2
. The two versions 
given below are equivalent and are 1 interchangeably. 
(a) The dimensional formula of force is MLT
-2
. 
(b) The dimensions of force are 1 in mass, 1 in length and -2 in time. 
Thus dimensional formula of any physical quantity is that expression that represents 
how and which of the base quantities are included in that quantity. 
Mass ? M 
b. Length ? L 
c.   Time ? T 
Temp. ? ?? , K 
Current ? I, A 
f. Amount of substance ? Mol 
Linear Momentum p = M × V 
? kg· m - sec
-1
 
= [M
1
 L
1
 T
-1
] 
Kinetic energy K =
1
2
MV
2
 
= M
1
× M
0
 L
1
 T
-1
× M
0
 L
1
 T
-1
 
= ?? 1
?? 2
?? -2
 
 MKS CGS 
_. Newton 10
5
 Dyne 
 
Question. Power ( P) =
 Energy 
 Time 
=
 Work 
 Time 
=
kg-m
2
-sec
-2
sec
 
 =
?? 1
?? 2
?? -2
?? 1
= ???? - ?? 2
- ?? -3
 = [?? 1
?? 2
?? -3
]
 
Question :Torque ?? ( Tau ) = Force × Distance 
 = kg- m - sec
-2
× m
 = kg- m
2
- sec
-2
 = [M
1
 L
2
 T
-2
]
 
Question Angular Momentum (L) = M × V × r 
 = ???? - ?? - ???? ?? -1
- ?? Unit Dimension  = ???? - ?? 2
- ?? -1
 = [M
1
 L
2
 T
-1
]
 
Question Coefficient of Viscosity ( ?? ) neta =
?? 6?????? 
?? = ?? ?? ??? ??? ,
?? =
?? ??? ?? ??? =
???? 
-2
?? 2
?? -1
?? =
???? ?? -2
?? ?? 2
?? ?? -1
= ???? - ?? -1
- sec
-1
?? = ?? ?? -1
?? -1
 = [?? 1
?? -1
?? -1
]
 
Question Pressure ( P) =
 Force 
 Area 
 
=
M
1
 L
1
 T
-2
 L
2
 
Dime. of P = M
1
 L
-1
 T
-2
 
 Unit = kg- m
-1
- sec
-2
= 1 pascal  
Question Surface Tension ( ?? ) =
 Force 
 Length 
 
=
?? 1
?? 1
?? -2
?? 
Dime. of ( ?? ) = ?? 1
?? 1
?? -2
 
 Unit = ???? - sec
-2
 
Question Universal Gravitational Constant (G)