Law of Conservation of Angular Momentum
The law of conservation of angular momentum states that “When the net external torque acting on a system about a given axis is zero, the total angular momentum of the system about that axis remains conserved.”
Mathematically, If Σ \(\vec{\tau}\) = 0 then \(\vec{J}\)= constant
Proof : According to the second law of motion, net force acting on a body is equal to its change of linear momentum i. e., \(\vec{F}=\frac{d \vec{P}}{d t}\)
Taking vector product of 
on both the sides of above expression
\(\vec{r} \times \vec{F}=\vec{r} \times \frac{\overrightarrow{d P}}{d t}\)
But \(\vec{r} \times \vec{F}\) is the torque acting on the body
∴ \(\vec{\tau}=\vec{r} \times \frac{\overrightarrow{d P}}{d t}\) …………….. (1)
Now angular momentum is defined as
\(\vec{J}=\vec{r} \times \vec{P}\)
Differentiating both the sides with respect to time (t)’
Which is the required equation.
This expression states that the torque acting on a particle is the time rate of change of its angular momentum. If the net external torque on the particle is zero, then,
\(\frac{d \overline{J}}{d t}\) = 0 or \(d \vec{J}\) = 0
Integrating both the sides
\(\vec{J}\) = constant
Thus the angular momentum of a particle is conserved if the net external torque acting on the particle is zero.
\(\vec{J}\) = Iω = constant
or I ∝ \(\frac{1}{\omega}\)
A system may consist of many number of particles or bodies. In case, there is a single non-rigid body, then during rotation, its moment of inertia may vary due to the change of distribution of mass about the axis of rotation. Therefore, in such a case (a single non rigid body), if moment of inertia changes from I1 to I2, then angular velocity must change from ω1 to ω2, so that
I1ω1 = I2ω2
Applications of Law of Conservation of Angular Momentum
Following are examples of some observed physical phenomena, which can be explained on the basis of the law of conservation of angular momentum :
(1) The angular velocity of a planet revolving in an elliptical orbit around the sun increases, when it comes near the sun and vice-versa : When the planet moving along its elliptical orbit is near the sun, its moment of inertia about the axis through the sun decreases and therefore its angular speed increases. On the other hand, when it is far away from the sun, its moment of inertia increases and hence angular speed decreases.
(2) A circus acrobat performs feats involving spin by bringing his arms and legs closer to his body or vice-versa. On bringing the arms and legs closer to the body, his moment of inertia / decreases. Hence to increases.
