Collision Theory
Linear Momentum:
If a particle of mass ‘m’ is moving with
velocity ‘ ⃗v’ then its linear momentum is denoted by
P=m⃗v with S.I. unit N.sec or kg.m/sec.
⃗
Angular Momentum:
The moment of linear momentum ‘‘ ⃗P’’
is called angular momentum & it is denoted by ⃗L
where ⃗L=⃗r ×⃗p=r⃗ ×m ⃗v .
Torque / Moment of force:
The turning effect of force is torque or
moment of force. It is denoted as τ⃗ ∨⃗H where
τ⃗ =⃗ F .
H =⃗r × ⃗
Force :
The tendency to produce or to stop motion of a
particle is known as its forces. It is denoted by ⃗F
Where ⃗F =ma⃗
Principle of Angular Momentum :
The time rate of change of angular momentum
of a particle about a pt. is equal to torque of the
, particle about the same pt. of the force acting on that
particle , known as Principal of angular momentum.
Proof :
We know by definition of angular momentum , ⃗L=⃗r × ⃗p
where ⃗P=m⃗v being linear momentum of the particle .
Now,
Differentiating w. r. t. ‘t’
d⃗
L d d ⃗r d
= ( r⃗ ×m ⃗v )= × m ⃗v + ⃗r × ( m⃗v )
dt dt dt dt
d ⃗v
=( ⃗v × m⃗v )+r⃗ × (m dt )
¿ m ( ⃗v ×⃗v ) +⃗r × ma⃗
=0 +r⃗ ×ma⃗
=r⃗ × ⃗F
=⃗H (Torque)
So, Time rate of change of Angular momentum.
Newton’s law of Restitution :
This law states that , ‘when two bodies collide ,
their relative velocity along the common normal, after
collision , bears a constant ratio ‘-e’ to the relative
velocity before collision. This law is called ‘‘Newton’s
law of Restitution’’.
Proof:
Consider two bodies m1,m2 having velocities u1,u2
before collision, and v1 , v2 be after collision.
Linear Momentum:
If a particle of mass ‘m’ is moving with
velocity ‘ ⃗v’ then its linear momentum is denoted by
P=m⃗v with S.I. unit N.sec or kg.m/sec.
⃗
Angular Momentum:
The moment of linear momentum ‘‘ ⃗P’’
is called angular momentum & it is denoted by ⃗L
where ⃗L=⃗r ×⃗p=r⃗ ×m ⃗v .
Torque / Moment of force:
The turning effect of force is torque or
moment of force. It is denoted as τ⃗ ∨⃗H where
τ⃗ =⃗ F .
H =⃗r × ⃗
Force :
The tendency to produce or to stop motion of a
particle is known as its forces. It is denoted by ⃗F
Where ⃗F =ma⃗
Principle of Angular Momentum :
The time rate of change of angular momentum
of a particle about a pt. is equal to torque of the
, particle about the same pt. of the force acting on that
particle , known as Principal of angular momentum.
Proof :
We know by definition of angular momentum , ⃗L=⃗r × ⃗p
where ⃗P=m⃗v being linear momentum of the particle .
Now,
Differentiating w. r. t. ‘t’
d⃗
L d d ⃗r d
= ( r⃗ ×m ⃗v )= × m ⃗v + ⃗r × ( m⃗v )
dt dt dt dt
d ⃗v
=( ⃗v × m⃗v )+r⃗ × (m dt )
¿ m ( ⃗v ×⃗v ) +⃗r × ma⃗
=0 +r⃗ ×ma⃗
=r⃗ × ⃗F
=⃗H (Torque)
So, Time rate of change of Angular momentum.
Newton’s law of Restitution :
This law states that , ‘when two bodies collide ,
their relative velocity along the common normal, after
collision , bears a constant ratio ‘-e’ to the relative
velocity before collision. This law is called ‘‘Newton’s
law of Restitution’’.
Proof:
Consider two bodies m1,m2 having velocities u1,u2
before collision, and v1 , v2 be after collision.
