Law of Conservation of Angular Momentum – statement and derivation

In this post, we will study the Law of Conservation of Angular Momentum with its 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 | equation derivation

This law can be mathematically derived very easily using one of the Torque equations.

Torque can be defined as the time rate of change of angular momentum.
τ = ΔL/ΔT …………(1)
Here, τ is the torque, ΔL is the change in angular momentum and ΔT is the time duration when this change happens.

For the situation in which the net torque τ is zero, ΔL/ΔT =0
This means when net torque is zero, the rate of change of angular momentum is also zero.
Therefore, we can say that when net torque is zero, the angular momentum (L) of a system remains constant.

L = constant, (when net τ = 0) ……………. (2)
This is an expression for the law of conservation of angular momentum.

As, L = Iω  (I: rotational inertia, ω: angular velocity), hence we can also write:
Iω = constant, (when net τ = 0) ……………. (3)
This is the second expression for the law of conservation of angular momentum.

Expanding equation 3 we get,
Iiωi = Ifωf (when net τ = 0) ……………. (4)
[Here, i denotes the initial condition, and f is for the final condition.]
This is the third expression for the law of conservation of angular momentum.

Thus we can complete the derivation of the formulas for the law of conservation of angular momentum.

Example and Implications – Ice skaters executing a spin

Ice skaters apply this principle skillfully. An example of conservation of angular momentum is seen in an ice skater executing a spin, as shown in.

Ice skaters apply the principle of conservation of angular momentum while executing a spin
Ice skaters apply the principle of conservation of angular momentum while executing a spin

The net torque on her is very close to zero because 1) there is relatively little friction between her skates and the ice, and 2) the friction is exerted very close to the pivot point. (Both F and r are small, and so Torque τ=r×F is negligibly small. )

When the ice skaters wish to spin rapidly, they wrap their arms tightly around their bodies. This results in the reduction of the moment of inertia. (I). To retain the total angular momentum constant, the angular velocity (ω) of the skater rises. That means, they decrease their moment of inertia to increase their angular velocity.

conclusion

Angular momentum remains constant if the net external torque applied on a system is zero. We can derive the expressions of 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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