Principle (or law) of conservation of angular momentum : The angular momentum of a body is conserved if the resultant external torque on the body is zero.
Proof : Consider a moving particle of mass m whose position vector with respect to the origin at any instant is \(\vec r\).
Then, at this instant, the linear velocity of this particle is \(\vec v\) = \(\frac{\vec{dr}}{dt}\), its linear momentum is \(\vec v\) = \(m\vec v\) and its angular momentum about an axis through the origin is \(\vec l\) = \(\vec r\) x \(\vec p\).
Suppose its angular momentum \(\vec l\) changes with time due to a torque \(\vec \tau\) exerted on the particle.
The time rate of change of its angular momentum,
∴ \(\vec l\) = constant, i.e., \(\vec l\) is conserved. This proves the principle (or law) of conservation of angular momentum.
Alternate Proof : Consider a rigid body rotating with angular acceleration \(\vec\alpha\) about the axis of rotation. If I is the moment of inertia of the body about the axis of rotation, \(\vec\omega\) the angular velocity of the body at time t and \(\vec L\) the corresponding angular momentum of the body, then
\(\vec L\) = \(I\vec\omega\) ....(1)
Differentiating this with respect to t, we get :
Rate of change of angular momentum with time,
( \(\because\) I = constant in a particular case, i.e. about the given axis of rotation)
But \(\vec\tau_{ext}\) = \(I\vec\propto\), where \(\vec\tau_{ext}\) is the resultant external torque on the body.
∴ \(\vec L\) = constant, i.e., \(\vec L\) is conserved. This proves the principle (or law) of conservation of angular momentum.
