# Law of Conservation of Angular Momentum – statement and derivation

## State the Law of Conservation of Angular Momentum

The Law of Conservation of Angular Momentum states that angular momentum remains constant if the net external torque applied on a system is zero.
So, when net external torque is zero on a body, then the net change in the angular momentum of the body is zero.

## Derive the expression for the Law of Conservation of Angular Momentum

This law can be mathematically derived very easily using one of the Torque equations,
We know that Torque = T = I α  ……………(1)

[ Torque is the product of Moment of Inertia (I) and α (alpha, which is angular acceleration) ]

Expanding the equation, we get
T = I (ω2-ω1)/t

[ here α  = angular acceleration
= time rate of change of angular velocity
= (ω2-ω1)/t
where ω2 and ω1 are final and initial angular velocities and t is the time gap]

or, T t =  I (ω2-ω1) ……………………(2)

**Torque is presented with the help of symbol τ (tao) or T.
From equation (2):
when, T = 0 (i.e., net torque is zero), then from the above equation we get,
I (ω2-ω1)  = 0
i.e., I ω2=I ω1 …………..  (3)

Iω2 represents final angular momentum and Iω1 represents initial angular momentum.
So, this shows that when net torque on a body is zero, then the angular momentum of the body remains unchanged. Thus we can do the derivation of the expression of the law.

## conclusion

Angular momentum remains constant if the net external torque applied on a system is zero. We can derive its expression and prove the law of Conservation of Angular Momentum mathematically with the help of a torque equation.

Law of Conservation of Angular Momentum – statement and derivation
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