Law of Conservation of Angular Momentum – statement and derivation

Statement of 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.

Derivation of 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 mathematically with the help of a torque equation.

Law of Conservation of Angular Momentum – statement and derivation
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