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Inertia:

Force: An external agent that try to change the state of a body.

Newton’s law of motion:

First Law:

Second Law:

Third Law:

Impulse:

Momentum:

Conservation of linear momentum:

When the net external force acting on a system is zero, then the total linear momentum of the system remains
constant.
Let us consider a body of mass m1 moves initially with velocity u1 and momentum P1i =m1u1. It
collides with a body of mass m2 moving initially with velocity u2 and momentum P2i =m2u2.
The two bodies are isolated from its environment, so that no forces act on either body during the collision
except the impulsive force that each body exert on the other.
After collision m1 moves with velocity v1 and momentum P1f = m1v1 and m2 with velocity v2 and momentum
P2f = m2v2
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At any particular time, the total momentum of the system consisting of the two bodies is
P = P1 + P2 ……………(1)
Differentiating equation (1) with respect to t
dP dP1 dP2
= + =  F1 +  F2
dt dt dt
Where, F1 and F2 are the net force acting on the body 1 and 2 respectively.
As the system is isolated before the collision no forces act on the bodies. So
F1 = 0
and F2 = 0
dP
And therefore, =0
dt
During the collision the only force acting on body 1 is F12, which is due to body 2. Similarly the only force
acting on body 2 is F21, which is due to body1.

F12 and F21 form an action reaction pair, so F12 = -F21 and
F12 + F21 = 0
dP
Thus, = 0 during collision too.
dt
dP
So at all times = 0 ……..(3)
dt
And P = constant
In other words, m1u1 + m2u2 = m1v1 + m2v2
That is the total momentum before the collision must be the same in magnitude and direction as the total
momentum after the collision. Even though P1 and P2 may both change as a result of collision their vector
sum stays the same.
Principle of conservation of energy:
Energy may be transformed from one kind to another, but it cannot be created or destroyed; the total energy
is constant. In other words, the total energy----- kinetic plus potential plus heat plus all other forms------
does not change, i.e.
K +  U + Q (change in other forms of energy ) = 0
Where, K = Change in K.E.
U = Sum of change in P.E.
Q = Heat energy
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Collisions:
In collision a relatively large force acts on each colliding particle for a relatively short time.
Properties of elastic collisions
(i) The total momentum of the colliding objects is always conserved.
(ii) The total kinetic energy is conserved.

Before collision After collision

m1 m2 m1 m2

u1 u2 v1 v2

Fig.1

Consider the head on elastic collision of two objects in one dimension as shown in Fig.1. The direction to
the right is positive.
Let us consider an object of mass m1 and velocity u1 colliding directly with another body of mass m2 and
velocity u2. After collision the velocities of m1 and m2 are m2 are v1 and v2 respectively.
According to conservation of momentum
m1u1 + m2u2 = m1v1 + m2v2 ……………(1)
Rearranging
m1 (u1 − v1 ) = m2 (v2 − u2 ) ………………(2)
Since the collision is elastic kinetic energy of the system is conserved. Therefore
1 2 1 2 1 2 1 2
m1u1 + m2u2 = m1v1 + m2v2 ……(3)
2 2 2 2
Rearranging
( 2 2
m1 u1 − v1 = m2 v2 − u2) ( 2 2
) ……….(4)

Dividing equ.(4) by equ.(2)

( 2
m1 u1 − v1
2
)
m v − u2
= 2 2
( 2 2
)
m1 (u1 − v1 ) m2 (u2 − v2 )

Or,
(u1 − v1 )(u1 + v1 ) = (v2 − u2 )(v2 + u2 )
(u1 − v1 ) (v2 − u2 )
Or, (u1 + v1 ) = (v2 + u2 )

Or, (u1 − u2 ) = (v2 − v1 ) …..……….(5)

Equation (5) tells us that relative speed of approach is equal to the relative speed of separation.
Now from equ.(5)
v2 = u1 − u2 + v1
Substituting the value of v2 in equ.(2)
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m1 (u1 − v1 ) = m2 (u1 − u2 + v1 − u2 )
= m2 (u1 − 2u2 + v1 )
Or, (m1 − m2 )u1 + 2m2u2 = (m1 + m2 )v1
(m − m2 )u1 2m2u2
Or, v1 = 1 + …………..(6)
(m1 + m2 ) (m1 + m2 )
Similarly,
2m1u1 (m − m1 )u2
v2 = + 2 ……………....(7)
(m1 + m2 ) (m1 + m2 )
Case I : (m1  m2 ) and u2 = 0 , i.e. massive target at rest
From equ.(6)
v1 = −u1
When a very small mass collides with a massive object the small mass will rebound in the opposite direction
with almost the same speed as its initial speed.

Case II : (m1 = m2 ) and u2 = 0 , i.e. equal masses and target at rest

From equ.(6)
v1 = 0
When a mass collides with another identical mass at rest a complete transfer of velocity will occur. m1 will
be stopped in its track and its velocity will be transferred completely to mass m2.
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Conservative and non-conservative forces and system:

Conservative Force: A conservative force is a force with the property that the work done in moving a
particle between two points is independent of the path taken. Equivalently if a particle travels in a closed
loop, the net work done by a conservative force is zero.
Suppose a particle starts at point A, and there is constant force acting on it. Then the particle is moved
around by other forces and eventually ends up at A again. Though the particle may still be moving, at that
instant when it passes A again, it has traveled a closed path. If the work done by the force F at this point is
zero then F passes the closed path test. Any force that passes closed path test is called conservative force.
For a conservative force work done by the force is stored in the system.
When a conservative force exists, it conserves mechanical energy.
Following equations are hold for conservative force

Following equations are hold for conservative force

 
(i)  F = 0

 
W =  F .dr = 0
(ii) c
 
(iii) F = − 

Gravitational force, elastic force, magnetic force etc. are example of conservative force.

Non-conservative Force: A non-conservative force is a force with the property that the work done in
moving a particle between two points depends on the path taken. Non conservative forces do not store
energy and it is called dissipative force. The energy that it removes from the system is no longer available
to the system for kinetic energy. Usually the energy is turned into heat, for example heat generated by
friction. In addition to heat friction also produces some sound energy. The water drag on a moving boat
converts the boat mechanical energy into not only heat and sound energy but also wave energy at the edges
of its wake.
Frictional force, air drag etc. are example of non conservative force.

Conservation of energy:
Conservation of energy principle states that the total amount of energy in an isolated system remains
constant over time. For an isolate system this law means that energy can change its location within the
system and that it can change its form within the system, for instance chemical energy can become kinetic
energy, but that energy can be neither created nor destroyed.
Conservation of momentum during collision:
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Let us consider a collision between two particles such as those of masses m1 and m2 as shown in figure.
During the brief collision exert large forces on each other. At any instant F1 is the force exerted on particle
1 by particle 2 and F2 is the force exerted on particle 2 by particle 1.
By Newtons third law these forces at any instant are equal in magnitude but oppositely directed.
The change in momentum of particle 1 resulting from the collision is
…………..(1)
In which is the average value of the force during the time interval of the collision
The change in momentum of particle 2 resulting from the collision is
…………..(1)
In which is the average value of the force during the time interval of the collision
If no other forces act on the particles, then and give the total change in momentum for each particle.
But we have seen that at each instant
So
If we consider the two particles as an isolated system, the total momentum of the system is

And the change in momentum of the system as a result of the collision is

Hence if there are no external forces the total momentum of the system is not changed by the collision. In
other words the total momentum of the system remains constant.

Center of mass:

Let us consider the simple case of a system of two particles m1 and m2 at a distance x1 and x2 respectively
from the origin O. Let the center of mass of the system is at a distance xCM from the origin.
According to the definition of center of mass the product of the total mass of the system and the distance
of the center of mass from the origin is equal to the sum of the products of the mass of each particle by its
distances fro the origin. i.e.
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……………….(1)
If we have n particles, m1, m2, …….mn, along a straight line, by definition the center of mass of these
particles relative to some origin is

……(2)
Where x1, x2, …..xn are the distances of the masses from the origin from which xCM is measured and is
the total mass of the system.
For a large number of particles lying In a plane, the center of mass the center of mass is at xCM, yCM,
where,
and

For a large number of particles not necessarily confined to a plane but distributed in space, the center of
mass is at xCM, yCM, zCM, where,

, and
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