Mechanics | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
MECHANICS 
1. INTRODUCTION 
Classical Mechanics, a branch of physics, deals with the motion of material bodies 
based on Newton's three laws:  
? the law of inertia 
? the law of force  
? the law of action and reaction 
which are fundamental concepts of nature. 
 
We shall first develop the structure of classical mechanics and then deal with its 
applications. 
2. A BRIEF SURVEY OF THE ELEMENTARY PRINCIPLES 
2.1 Mechanics of a particle 
Newton's second law of motion states that the force acting on a particle equals the rate 
of change in its linear momentum, with the change occurring in the direction of the force. 
If ???
 is the total force acting on the parle and  ??? is the linear momentum, then 
???
=
????
?????
????
=
?? ????
(?? ???) 
???
=
?? ????
(?? ????
?????
????
)=?? ?? 2
???
?? ?? 2
=?? ??? 
The ?? is the mass of the particle and ???, the acceleration of the particle (Here ??? denotes 
position vector of the particle with respect to a fixed point of reference). Newton's second 
law of motion defines mass and force by assuming that the particle's mass remains 
constant over time. 
Conservation theorems in mechanics provide crucial conclusions and observations 
about certain mechanical quantities, indicating under what conditions these various 
physical quantities are constants in time. 
 
 
(a) Law (Theorem) of conservation of the Linear momentum of a particle: 
If the total force ???
 is zero on a particle, then its linear momentum ??? is conserved. 
(Remark: We prefer to call it a theorem than a law; for we can prove it using Newton's 
laws of motion. Usually, laws are accepted facts and are not proved)  
Page 2


Edurev123 
MECHANICS 
1. INTRODUCTION 
Classical Mechanics, a branch of physics, deals with the motion of material bodies 
based on Newton's three laws:  
? the law of inertia 
? the law of force  
? the law of action and reaction 
which are fundamental concepts of nature. 
 
We shall first develop the structure of classical mechanics and then deal with its 
applications. 
2. A BRIEF SURVEY OF THE ELEMENTARY PRINCIPLES 
2.1 Mechanics of a particle 
Newton's second law of motion states that the force acting on a particle equals the rate 
of change in its linear momentum, with the change occurring in the direction of the force. 
If ???
 is the total force acting on the parle and  ??? is the linear momentum, then 
???
=
????
?????
????
=
?? ????
(?? ???) 
???
=
?? ????
(?? ????
?????
????
)=?? ?? 2
???
?? ?? 2
=?? ??? 
The ?? is the mass of the particle and ???, the acceleration of the particle (Here ??? denotes 
position vector of the particle with respect to a fixed point of reference). Newton's second 
law of motion defines mass and force by assuming that the particle's mass remains 
constant over time. 
Conservation theorems in mechanics provide crucial conclusions and observations 
about certain mechanical quantities, indicating under what conditions these various 
physical quantities are constants in time. 
 
 
(a) Law (Theorem) of conservation of the Linear momentum of a particle: 
If the total force ???
 is zero on a particle, then its linear momentum ??? is conserved. 
(Remark: We prefer to call it a theorem than a law; for we can prove it using Newton's 
laws of motion. Usually, laws are accepted facts and are not proved)  
Proof: 
We have 0=???
=
????
??????
????
 (Newton's second law). This implies that ??? is a constant with 
respect to time, or in other words the linear momeritum ??? is conserved.  
Definition 
The angular momentum ????
 of a particle about a point ?? is defined as the moment of 
momentum about ?? . i.e. ????
=???×???  where ??? is the position vector of the particle ?? . 
 
(b) Law (Theorem) of conservation of angular momentum of a particle: 
If the total torque or the moment of force ???
 on a particle is zero, then its angular 
momentum ????
 is conserved. 
 
Proof: 
By definition, the total torque ?????
 about ?? is the moment of force ???
 about ?? , where ???
 is the 
total force on the particle. Therefore, if ?????
 is zero then moment of force ???
 about ?? is zero. 
i.e.  
0=?????
=???×???
=???×
?? ?? ?? (?? ???)=???×
?? ????
(?? ?? ???
????
) 
Now 
?? ????
(???×?? ???)=(
?? ???
????
×?? ???)+(???×
?? ????
(?? ???))=(???×?? ???)+(???×
?? ????
(?? ???)) 
=0+(???×
?? ????
(?? ???))=???×
?? ????
(?? ???)…(1) 
 
Thus ?????
=0?0=???×
?? ????
(?? ???)=
?? ????
(???×?? ???) 
=
?? ????
(???×???)=
????
?????
????
 on using (1).  
d
???
?? ?? t
=0 or L
??
 is conserved 
Kinetic Energy of a particle 
By energy we mean the capacity for doing work against resistance. The work done by 
the external force ???
 upon a particle in moving it from position (1) to position (2) is  
Page 3


Edurev123 
MECHANICS 
1. INTRODUCTION 
Classical Mechanics, a branch of physics, deals with the motion of material bodies 
based on Newton's three laws:  
? the law of inertia 
? the law of force  
? the law of action and reaction 
which are fundamental concepts of nature. 
 
We shall first develop the structure of classical mechanics and then deal with its 
applications. 
2. A BRIEF SURVEY OF THE ELEMENTARY PRINCIPLES 
2.1 Mechanics of a particle 
Newton's second law of motion states that the force acting on a particle equals the rate 
of change in its linear momentum, with the change occurring in the direction of the force. 
If ???
 is the total force acting on the parle and  ??? is the linear momentum, then 
???
=
????
?????
????
=
?? ????
(?? ???) 
???
=
?? ????
(?? ????
?????
????
)=?? ?? 2
???
?? ?? 2
=?? ??? 
The ?? is the mass of the particle and ???, the acceleration of the particle (Here ??? denotes 
position vector of the particle with respect to a fixed point of reference). Newton's second 
law of motion defines mass and force by assuming that the particle's mass remains 
constant over time. 
Conservation theorems in mechanics provide crucial conclusions and observations 
about certain mechanical quantities, indicating under what conditions these various 
physical quantities are constants in time. 
 
 
(a) Law (Theorem) of conservation of the Linear momentum of a particle: 
If the total force ???
 is zero on a particle, then its linear momentum ??? is conserved. 
(Remark: We prefer to call it a theorem than a law; for we can prove it using Newton's 
laws of motion. Usually, laws are accepted facts and are not proved)  
Proof: 
We have 0=???
=
????
??????
????
 (Newton's second law). This implies that ??? is a constant with 
respect to time, or in other words the linear momeritum ??? is conserved.  
Definition 
The angular momentum ????
 of a particle about a point ?? is defined as the moment of 
momentum about ?? . i.e. ????
=???×???  where ??? is the position vector of the particle ?? . 
 
(b) Law (Theorem) of conservation of angular momentum of a particle: 
If the total torque or the moment of force ???
 on a particle is zero, then its angular 
momentum ????
 is conserved. 
 
Proof: 
By definition, the total torque ?????
 about ?? is the moment of force ???
 about ?? , where ???
 is the 
total force on the particle. Therefore, if ?????
 is zero then moment of force ???
 about ?? is zero. 
i.e.  
0=?????
=???×???
=???×
?? ?? ?? (?? ???)=???×
?? ????
(?? ?? ???
????
) 
Now 
?? ????
(???×?? ???)=(
?? ???
????
×?? ???)+(???×
?? ????
(?? ???))=(???×?? ???)+(???×
?? ????
(?? ???)) 
=0+(???×
?? ????
(?? ???))=???×
?? ????
(?? ???)…(1) 
 
Thus ?????
=0?0=???×
?? ????
(?? ???)=
?? ????
(???×?? ???) 
=
?? ????
(???×???)=
????
?????
????
 on using (1).  
d
???
?? ?? t
=0 or L
??
 is conserved 
Kinetic Energy of a particle 
By energy we mean the capacity for doing work against resistance. The work done by 
the external force ???
 upon a particle in moving it from position (1) to position (2) is  
?? 12
=?
(1)
????
?? ???????? m
d
dr ? 
Since,  Force = mass × acceleration 
Thus ?? 12
=?? ?
(1)
(2)
?
????
????
????
????
???? (as mass ?? is assumed to be constant) 
 =?? ? ?
(2)
(1)
?
?? ???
????
·????
 =?? ? ?
(2)
(1)
?
1
2
?? ????
(???·???)
 =
1
2
?? ? ?
(2)
(1)
?
?? ????
(?? ?? 2
)????
 =
?? 2
(?? 2
2
-?? 1
2
)
 
where ?? 1
 and ?? 2
 are the magnitudes of the velocity at (1) and (2) respectively. If ?? 1
 and 
?? 2
 are the kinetic energies of the particle at (1) and (2) respectively. We have ?? 12
=?? 2
-
?? 1
. [Recall that the scalar quantity 
1
2
?? ?? 2
 is by definition the kinetic energy of the particle]. 
In other words, the work done by the external force ?????
 on particle ?? in moving it 
from position (1) to position (2) is equal to the change in the kinetic energy. 
Definition:  
If the work done by a force field ???
 in moving a particle (a system of particles) around any 
closed orbit in a domain D is zero, then the force field (and the system) is said io be 
conservative. i.e. ???
 is a conservative force field in a domain ?? if ?? ???
·?? ???=0 around 
every closed path lying with in D. 
Note: 
(1) ???
 is conservative if and only if ?×???
= curl ???
=?? . This follows immediately from 
Stoke's theorem. In fact, by Stoke's theorem, ? 
?? ???
·?? ???= ?
?? ??(?×???
)=??ˆ???? (with the 
usual notations). Therefore, ???
 is conservative, ?
?? ??(?×???
)·??ˆ???? is zero for any arbitrary 
surface ?? with in the domain of ???
 and hence ?×???
=0 in its domain. The converse is 
quite clear. 
(2) ???
 is conservative in a domain ?? if there exists a scaler point function ?? such that ???
=
-??? (Such a function ?? is called a potential function of ???
 or the potentiai energy of the 
system). This follows from our preceding observation (1) which states that ???
 is 
conservative ?.?×???
=0; which in turn is true if ???
 is centre form -??? for some scalar 
point function ?? a standard result in Vector analysis. 
(3) If a force field is conservative, the work done by it in moving a particle from an initial 
position (1) to a final position (2) is independent of the path of integration between (1) 
and (2). Suppose ?? 1
 and ?? 2
 are two paths joining "(1) and (2) then C=C
1
-C
2
 (with 
proper orientalon) is closed path and ???
 being cons rrvative, we have  
Page 4


Edurev123 
MECHANICS 
1. INTRODUCTION 
Classical Mechanics, a branch of physics, deals with the motion of material bodies 
based on Newton's three laws:  
? the law of inertia 
? the law of force  
? the law of action and reaction 
which are fundamental concepts of nature. 
 
We shall first develop the structure of classical mechanics and then deal with its 
applications. 
2. A BRIEF SURVEY OF THE ELEMENTARY PRINCIPLES 
2.1 Mechanics of a particle 
Newton's second law of motion states that the force acting on a particle equals the rate 
of change in its linear momentum, with the change occurring in the direction of the force. 
If ???
 is the total force acting on the parle and  ??? is the linear momentum, then 
???
=
????
?????
????
=
?? ????
(?? ???) 
???
=
?? ????
(?? ????
?????
????
)=?? ?? 2
???
?? ?? 2
=?? ??? 
The ?? is the mass of the particle and ???, the acceleration of the particle (Here ??? denotes 
position vector of the particle with respect to a fixed point of reference). Newton's second 
law of motion defines mass and force by assuming that the particle's mass remains 
constant over time. 
Conservation theorems in mechanics provide crucial conclusions and observations 
about certain mechanical quantities, indicating under what conditions these various 
physical quantities are constants in time. 
 
 
(a) Law (Theorem) of conservation of the Linear momentum of a particle: 
If the total force ???
 is zero on a particle, then its linear momentum ??? is conserved. 
(Remark: We prefer to call it a theorem than a law; for we can prove it using Newton's 
laws of motion. Usually, laws are accepted facts and are not proved)  
Proof: 
We have 0=???
=
????
??????
????
 (Newton's second law). This implies that ??? is a constant with 
respect to time, or in other words the linear momeritum ??? is conserved.  
Definition 
The angular momentum ????
 of a particle about a point ?? is defined as the moment of 
momentum about ?? . i.e. ????
=???×???  where ??? is the position vector of the particle ?? . 
 
(b) Law (Theorem) of conservation of angular momentum of a particle: 
If the total torque or the moment of force ???
 on a particle is zero, then its angular 
momentum ????
 is conserved. 
 
Proof: 
By definition, the total torque ?????
 about ?? is the moment of force ???
 about ?? , where ???
 is the 
total force on the particle. Therefore, if ?????
 is zero then moment of force ???
 about ?? is zero. 
i.e.  
0=?????
=???×???
=???×
?? ?? ?? (?? ???)=???×
?? ????
(?? ?? ???
????
) 
Now 
?? ????
(???×?? ???)=(
?? ???
????
×?? ???)+(???×
?? ????
(?? ???))=(???×?? ???)+(???×
?? ????
(?? ???)) 
=0+(???×
?? ????
(?? ???))=???×
?? ????
(?? ???)…(1) 
 
Thus ?????
=0?0=???×
?? ????
(?? ???)=
?? ????
(???×?? ???) 
=
?? ????
(???×???)=
????
?????
????
 on using (1).  
d
???
?? ?? t
=0 or L
??
 is conserved 
Kinetic Energy of a particle 
By energy we mean the capacity for doing work against resistance. The work done by 
the external force ???
 upon a particle in moving it from position (1) to position (2) is  
?? 12
=?
(1)
????
?? ???????? m
d
dr ? 
Since,  Force = mass × acceleration 
Thus ?? 12
=?? ?
(1)
(2)
?
????
????
????
????
???? (as mass ?? is assumed to be constant) 
 =?? ? ?
(2)
(1)
?
?? ???
????
·????
 =?? ? ?
(2)
(1)
?
1
2
?? ????
(???·???)
 =
1
2
?? ? ?
(2)
(1)
?
?? ????
(?? ?? 2
)????
 =
?? 2
(?? 2
2
-?? 1
2
)
 
where ?? 1
 and ?? 2
 are the magnitudes of the velocity at (1) and (2) respectively. If ?? 1
 and 
?? 2
 are the kinetic energies of the particle at (1) and (2) respectively. We have ?? 12
=?? 2
-
?? 1
. [Recall that the scalar quantity 
1
2
?? ?? 2
 is by definition the kinetic energy of the particle]. 
In other words, the work done by the external force ?????
 on particle ?? in moving it 
from position (1) to position (2) is equal to the change in the kinetic energy. 
Definition:  
If the work done by a force field ???
 in moving a particle (a system of particles) around any 
closed orbit in a domain D is zero, then the force field (and the system) is said io be 
conservative. i.e. ???
 is a conservative force field in a domain ?? if ?? ???
·?? ???=0 around 
every closed path lying with in D. 
Note: 
(1) ???
 is conservative if and only if ?×???
= curl ???
=?? . This follows immediately from 
Stoke's theorem. In fact, by Stoke's theorem, ? 
?? ???
·?? ???= ?
?? ??(?×???
)=??ˆ???? (with the 
usual notations). Therefore, ???
 is conservative, ?
?? ??(?×???
)·??ˆ???? is zero for any arbitrary 
surface ?? with in the domain of ???
 and hence ?×???
=0 in its domain. The converse is 
quite clear. 
(2) ???
 is conservative in a domain ?? if there exists a scaler point function ?? such that ???
=
-??? (Such a function ?? is called a potential function of ???
 or the potentiai energy of the 
system). This follows from our preceding observation (1) which states that ???
 is 
conservative ?.?×???
=0; which in turn is true if ???
 is centre form -??? for some scalar 
point function ?? a standard result in Vector analysis. 
(3) If a force field is conservative, the work done by it in moving a particle from an initial 
position (1) to a final position (2) is independent of the path of integration between (1) 
and (2). Suppose ?? 1
 and ?? 2
 are two paths joining "(1) and (2) then C=C
1
-C
2
 (with 
proper orientalon) is closed path and ???
 being cons rrvative, we have  
0=?? ·???
·????
?????
=?? ???
·???? ·-????
·????  
 
This implies that  ?? 12
=?
(1)
(2)
????
·????
?????
 is independent of the path of integration joining (1) 
and (2). 
 
 
(c) The Law (Theorem) of conservation of energy 
If the forces acting on a particle are conservative then the total energy ?? =?? +?? of the 
particle is conserved.  
Proof:  
With the notations introduced either, we have ?? 12
=?
(1)
(2)
????
·????
?????
=?? 2
-?? 1
 
Also ?? 12
=?
(1)
(2)
????
·????
?????
=?
(1)
(2)
?(-??? )·????
?????
 where ?? is the potential function of the force field 
???
  
This gives  ?? 12
=?
(1)
(2)
?-???? =-(?? 2
-?? 1
) . 
Thus ?? 12
=?? 2
-?? 1
?-(?? 2
-?? 1
)=?? ?? -?? 2
 
This implies that  ?? 1
+?? 1
=?? 2
+?? 2
=Kinetic energy+ Potential energy  
Total energy is a constant with respect to time, whenever the force field is conservative. 
 
2.2 Mechanics of a system of particles 
We shall now extend the conservation theorems to a system of particles. Studying a 
system of particles requires distinguishing between external and internal forces, with 
external forces coming from external sources and internal forces from all system 
particles 
 The equation of motion as per Newton second law, for ith particle takes the form  
Page 5


Edurev123 
MECHANICS 
1. INTRODUCTION 
Classical Mechanics, a branch of physics, deals with the motion of material bodies 
based on Newton's three laws:  
? the law of inertia 
? the law of force  
? the law of action and reaction 
which are fundamental concepts of nature. 
 
We shall first develop the structure of classical mechanics and then deal with its 
applications. 
2. A BRIEF SURVEY OF THE ELEMENTARY PRINCIPLES 
2.1 Mechanics of a particle 
Newton's second law of motion states that the force acting on a particle equals the rate 
of change in its linear momentum, with the change occurring in the direction of the force. 
If ???
 is the total force acting on the parle and  ??? is the linear momentum, then 
???
=
????
?????
????
=
?? ????
(?? ???) 
???
=
?? ????
(?? ????
?????
????
)=?? ?? 2
???
?? ?? 2
=?? ??? 
The ?? is the mass of the particle and ???, the acceleration of the particle (Here ??? denotes 
position vector of the particle with respect to a fixed point of reference). Newton's second 
law of motion defines mass and force by assuming that the particle's mass remains 
constant over time. 
Conservation theorems in mechanics provide crucial conclusions and observations 
about certain mechanical quantities, indicating under what conditions these various 
physical quantities are constants in time. 
 
 
(a) Law (Theorem) of conservation of the Linear momentum of a particle: 
If the total force ???
 is zero on a particle, then its linear momentum ??? is conserved. 
(Remark: We prefer to call it a theorem than a law; for we can prove it using Newton's 
laws of motion. Usually, laws are accepted facts and are not proved)  
Proof: 
We have 0=???
=
????
??????
????
 (Newton's second law). This implies that ??? is a constant with 
respect to time, or in other words the linear momeritum ??? is conserved.  
Definition 
The angular momentum ????
 of a particle about a point ?? is defined as the moment of 
momentum about ?? . i.e. ????
=???×???  where ??? is the position vector of the particle ?? . 
 
(b) Law (Theorem) of conservation of angular momentum of a particle: 
If the total torque or the moment of force ???
 on a particle is zero, then its angular 
momentum ????
 is conserved. 
 
Proof: 
By definition, the total torque ?????
 about ?? is the moment of force ???
 about ?? , where ???
 is the 
total force on the particle. Therefore, if ?????
 is zero then moment of force ???
 about ?? is zero. 
i.e.  
0=?????
=???×???
=???×
?? ?? ?? (?? ???)=???×
?? ????
(?? ?? ???
????
) 
Now 
?? ????
(???×?? ???)=(
?? ???
????
×?? ???)+(???×
?? ????
(?? ???))=(???×?? ???)+(???×
?? ????
(?? ???)) 
=0+(???×
?? ????
(?? ???))=???×
?? ????
(?? ???)…(1) 
 
Thus ?????
=0?0=???×
?? ????
(?? ???)=
?? ????
(???×?? ???) 
=
?? ????
(???×???)=
????
?????
????
 on using (1).  
d
???
?? ?? t
=0 or L
??
 is conserved 
Kinetic Energy of a particle 
By energy we mean the capacity for doing work against resistance. The work done by 
the external force ???
 upon a particle in moving it from position (1) to position (2) is  
?? 12
=?
(1)
????
?? ???????? m
d
dr ? 
Since,  Force = mass × acceleration 
Thus ?? 12
=?? ?
(1)
(2)
?
????
????
????
????
???? (as mass ?? is assumed to be constant) 
 =?? ? ?
(2)
(1)
?
?? ???
????
·????
 =?? ? ?
(2)
(1)
?
1
2
?? ????
(???·???)
 =
1
2
?? ? ?
(2)
(1)
?
?? ????
(?? ?? 2
)????
 =
?? 2
(?? 2
2
-?? 1
2
)
 
where ?? 1
 and ?? 2
 are the magnitudes of the velocity at (1) and (2) respectively. If ?? 1
 and 
?? 2
 are the kinetic energies of the particle at (1) and (2) respectively. We have ?? 12
=?? 2
-
?? 1
. [Recall that the scalar quantity 
1
2
?? ?? 2
 is by definition the kinetic energy of the particle]. 
In other words, the work done by the external force ?????
 on particle ?? in moving it 
from position (1) to position (2) is equal to the change in the kinetic energy. 
Definition:  
If the work done by a force field ???
 in moving a particle (a system of particles) around any 
closed orbit in a domain D is zero, then the force field (and the system) is said io be 
conservative. i.e. ???
 is a conservative force field in a domain ?? if ?? ???
·?? ???=0 around 
every closed path lying with in D. 
Note: 
(1) ???
 is conservative if and only if ?×???
= curl ???
=?? . This follows immediately from 
Stoke's theorem. In fact, by Stoke's theorem, ? 
?? ???
·?? ???= ?
?? ??(?×???
)=??ˆ???? (with the 
usual notations). Therefore, ???
 is conservative, ?
?? ??(?×???
)·??ˆ???? is zero for any arbitrary 
surface ?? with in the domain of ???
 and hence ?×???
=0 in its domain. The converse is 
quite clear. 
(2) ???
 is conservative in a domain ?? if there exists a scaler point function ?? such that ???
=
-??? (Such a function ?? is called a potential function of ???
 or the potentiai energy of the 
system). This follows from our preceding observation (1) which states that ???
 is 
conservative ?.?×???
=0; which in turn is true if ???
 is centre form -??? for some scalar 
point function ?? a standard result in Vector analysis. 
(3) If a force field is conservative, the work done by it in moving a particle from an initial 
position (1) to a final position (2) is independent of the path of integration between (1) 
and (2). Suppose ?? 1
 and ?? 2
 are two paths joining "(1) and (2) then C=C
1
-C
2
 (with 
proper orientalon) is closed path and ???
 being cons rrvative, we have  
0=?? ·???
·????
?????
=?? ???
·???? ·-????
·????  
 
This implies that  ?? 12
=?
(1)
(2)
????
·????
?????
 is independent of the path of integration joining (1) 
and (2). 
 
 
(c) The Law (Theorem) of conservation of energy 
If the forces acting on a particle are conservative then the total energy ?? =?? +?? of the 
particle is conserved.  
Proof:  
With the notations introduced either, we have ?? 12
=?
(1)
(2)
????
·????
?????
=?? 2
-?? 1
 
Also ?? 12
=?
(1)
(2)
????
·????
?????
=?
(1)
(2)
?(-??? )·????
?????
 where ?? is the potential function of the force field 
???
  
This gives  ?? 12
=?
(1)
(2)
?-???? =-(?? 2
-?? 1
) . 
Thus ?? 12
=?? 2
-?? 1
?-(?? 2
-?? 1
)=?? ?? -?? 2
 
This implies that  ?? 1
+?? 1
=?? 2
+?? 2
=Kinetic energy+ Potential energy  
Total energy is a constant with respect to time, whenever the force field is conservative. 
 
2.2 Mechanics of a system of particles 
We shall now extend the conservation theorems to a system of particles. Studying a 
system of particles requires distinguishing between external and internal forces, with 
external forces coming from external sources and internal forces from all system 
particles 
 The equation of motion as per Newton second law, for ith particle takes the form  
 ???
?? = ???
?? =???
?? +?
?? ??? ????
????
, where ???
?? is the linear momentum of the ith particle. ???
?? is the total 
force on the ith particle,  ???
?? is the total external force on the ith particle and ???
????
 is the force 
on the ith particle due to the jth particle of the system. Again, by Newton's third law, for 
?? ??? we have ???
????
=-???
????
 
Summing over all particles, we get, 
? ?
i
???
1
=? ?
1
???
1
(?? )
+? ?
?? ?1
? ?
?? ???
?? 
i.e. ?
?? ?
?? 2
?? ?? 2
(?? ?? ???
?? )=
?? 2
?? ?? 2
(?
?? ??? ?? ???
?? ) 
=? ??????
?? (?? )
=???
(?? )
= total external force, since the second sum ?
????
??? ????
?????
=0.? 
The centre of mass of the system of particles, by definition, has the position vector 
????
=
? ?? ?? ???
?? ? ?? ?? =
? ?? ?? ???
?? ?? 
where ?? =? ?? ?? . Newton's second law applied to the system of particles gives, 
?? 2
?? ?? 2
(? ?
?? ?m
i
.???)=
?? 2
?? ?? 2
·(?? ????
) 
 ?? ?? 2
????
?? ?? 2
=???
(?? )= total external force 
 
The system's motion is influenced by net external forces acting on its total mass, 
implying that internal forces do not significantly impact its motion. 
(a) Conservation of linear momentum of a system of particles: 
If the total external force is zero on a system of particles; then the total linear momentum 
of the system is conserved. 
Proof:  
If ????
 denotes the total linear momentum of the system then  ???=S?? ?? ¨
???
?? =(? ?? ?? )
?? ????
(
? ?? ?? ???
?? S?? ?? ) 
i.e., ????
=?? ?? ????
????
 (usual notations). 
? total external force = 0=???
(?? )
=?? ?? 2
????
?? ?? 2
=
?? ????
????
 
This implies that if the total external ???
(?? )
=0 then the total linear momentum of the 
system is conserved.