This document discusses linear momentum, angular momentum, and torque. It defines linear momentum as the product of an object's mass and velocity. Angular momentum is defined as the vector product of the position vector from a fixed point to an object and the object's linear momentum. For a rigid body, the angular momentum is the sum of the angular momenta of all its particles about the axis of rotation, and equals the moment of inertia times the angular velocity. Torque is defined as the cross product of the position vector and applied force vector.

1. Linear and Angular Momentum
C.D. Mungmode
M. G. College, Armori

2. • Linear Momentum:
When particle with mass m moves with velocity
v, we define its Linear Momentum p as product
of its mass m and its velocity v:
Unit of linear momentum is kg m / s. There is no
special name for this unit.
It is the measure of how hard it is to stop or turn
a moving object.

3. • Conservation of Linear Momentum:
Consider a system of particles.
The total linear momentum of a system of particles is constant whenever
the vector sum of the external forces on the system is zero and subjected
to their mutual interaction. In particular, the total momentum of an
isolated system is constant.
If Σ Fext = 0, then ΔP = 0, so P = constant, and in any instant of time
P1 = P2

4. • Angular Momentum:
Consider a point-like particle of mass m moving with a velocity v.
The linear momentum of the particle is p = mv. Consider a point S located
anywhere in space. Let r denote the vector from the point S to the location
of the object.
Define the angular momentum J̅ about the point S of a point-like particle
as the vector product of the vector from the point S to the location of the
object with the linear momentum of the particle,
It is also defined as moment of its linear momentum about a fixed point.

5. The derived SI units for angular momentum are [kg ⋅m2 ⋅s−1]
= [N⋅m⋅s] = [J ⋅s]. There is no special name for this set of units.
It is a vector quantity. Its direction is perpendicular to both r
and p.
The magnitude of it can also be written as

6. • Angular Momentum of a Rigid Body:
The sum of the moments of the linear momentum of all the particles of a rotating rigid
body about the axis of rotation is called its angular momentum.
Consider the particles of mass m1, m2, ..... of the rigid body lying at distance r1, r2, ....
from the axis of rotation having linear velocities v1, v2, .... Respectively. ω is the
magnitude of angular velocity, then
Linear momentum of particle m1 = m1v1 = m1r1ω
Hence, angular momentum of particle m1 = m1v1r1 = m1r1
2ω
Similarly, for second particle of mass m2, angular momentum is = m2r2
2ω
Therefore, angular momentum of all the particles = m1r1
2ω + m2r2
2ω + .....
J = Σ mr2ω = Iω
Where, I = Σ mr2 is moment of inertia of rigid body.

7. • Torque:
If a force F acts on a particle at a point P whose position with respect to the origin O is
given by displacement vector r, the torque τ on the particle with respect to origin O is
defined as,
τ = r x F