6.5 Conservation of Angular Momentum

Conservation of Angular Momentum

When there are multiple rigid bodies rotating about the same axis, the total angular momentum of the system will just be the sum of all of the individual angular momentums of each body. Just like how linear momentum is conserved if there is no external net force, the same concept also applies to angular momentum. Conservation of angular momentum is a fundamental principle, stating that if no external torque acts on a system, the total angular momentum of that system remains constant.

From Newton’s 3rd Law in Rotational Form, we can get that if 2 rigid bodies in a system interact through an internal torque, they will exert the same torque but in opposite directions on each other for the same time interval that they are in contact. From the equation L=t, both objects will experience the same change in momentum but just in opposite directions. From this analysis, we can see that an internal torque in a system will not change the total angular momentum of the system.

Just like with conservation of linear momentum, a system can be chosen so that the angular momentum is conserved or not conserved. Let’s say you have 2 rigid bodies with the same rotational inertia, with 1 having an angular velocity of ${\omega}$ and the other having an angular velocity of 0. Then, the 2 rigid bodies collide, making their combined angular velocity ${\frac{\omega}{2}}$. From the system of both bodies, the angular momentum is conserved, as before it was ${L_{tot}=I \omega}$ and after it was ${L_{tot}=(2I)\frac{\omega}{2}}$. However, if you look at only the object that started with an angular velocity, its angular momentum was not conserved as before it was ${L_{tot}=I \omega}$ and after it was ${L_{tot}=I \frac{\omega}{2}}$.

Angular momentum is also conserved in scenarios where a body's rotational inertia changes. Imagine if a ball starts rotating about an axis through a string at a very long length. If the string is slowly reduced in length, this will decrease the ball’s rotational inertia, which is found with the equation ${I=mr^2}$. However, due to angular momentum also needing to be conserved, if the rotational inertia is decreased, then the ball’s angular velocity will increase to compensate.

In certain scenarios where angular momentum is not conserved, it can only be caused from a nonzero net external torque exerting an angular impulse on the system. If a system has a change in total angular momentum, it will be the same amount as the angular impulse exerted on the system. However, in a closed system, angular momentum is always conserved. Angular momentum is conserved in all interactions, it is just that in open systems, some of the angular momentum is transferred into or out of the system.\