Question

State and prove the law of conservation of angular momentum.

Hint:

This is the rotational equivalent of the conservation of linear momentum. No external force means linear momentum is constant. No external torque means angular momentum is constant. Think of an ice skater spinning faster when she pulls her arms in.

Answer 1

Statement: The law of conservation of angular momentum states that if the net external torque acting on a system is zero, then the total angular momentum of the system remains constant (conserved).
Proof: The angular momentum (\(\vec{L}\)) of a system is related to its moment of inertia (\(I\)) and angular velocity (\(\vec{\omega}\)) by \(\vec{L} = I\vec{\omega}\).
The relationship between torque (\(\vec{\tau}\)) and angular momentum (\(\vec{L}\)) is that the net external torque is equal to the time rate of change of the angular momentum.
\[ \vec{\tau}_{ext} = \frac{d\vec{L}}{dt} \] According to the law, if the net external torque on the system is zero, then \(\vec{\tau}_{ext} = 0\).
Substituting this into the equation: \[ 0 = \frac{d\vec{L}}{dt} \] This implies that \(\vec{L}\) is a constant with respect to time. The derivative of a constant is zero.
Therefore, \(\vec{L} = \text{constant}\).
This proves that the angular momentum of the system is conserved when no net external torque acts on it.