# Angular Momentum

## Angular Momentum Definition

A body possessing linear velocity has linear momentum. In the same way, a rotational body has rotational momentum called angular momentum.

Angular momentum is the moment of linear momentum. It defines the quantity of motion of rigid rotational bodies. Angular momentum is defined as the vector product( or cross product) of distance from the axis of rotation with the linear momentum. It is denoted by

In Vector form, It is written as

where is a unit vector along the direction of and is the angle between and

In magnitude:

Consider a rigid body rotating with uniform angular velocity ‘‘ and one particle of mass .

Angular momentum of that particle =

But and So, angular momentum becomes

If there are ‘n’ particles in a rigid body, the total angular momentum can be calculated by adding the angular momentum of all particles. So, Total angular momentum can be written as:

Differentiating with respect to time, we get:

Hence, the rate of change of angular momentum is called torque.

## Conservation of Angular Momentum

Conservation of Angular Momentum states that “If no external torque is acted on the system, the total angular momentum always remains constant.”

i.e. If then, = constant

We have

Differentiating with respect to t we get:

If then,

= constant I = constant

i.e.

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