Conservation of Angular Momentum

Conservation of Angular Momentum is a law of which states that the total angular momentum of a system, about an axis, remains constant, so long as no external unbalanced torque acts on the system, about that axis.

Equation of Law of Conservation of Angular Momentum

Mathematically, the law of conservation of angular momentum can be stated as:
If $\sum {\vec \tau} = 0$, then $\vec L = constant$.
When a system is rotating then,
$\frac{d \vec L}{dt} = \vec \tau$

This equation means that when a net torque acts on a rotating body, it changes the system’s angular momentum, and the time rate of change of angular momentum is equal to torque. This means that, if net torque is zero, then,
$\frac{d \vec L}{dt} = 0$
or
$d \vec L = 0$
Now Integrating the equation gives,
$\vec L = constant$
meaning the angular momentum of the system is constant.

Absolute Symmetry of Nature

This law of conservation of angular momentum is an absolute symmetry of nature. All objects in nature obeys this principle. This principle can be utilized to analyze orbits of planets, or satellites. We can study the properties of rotating bodies like moment of inertia, rotational kinetic energy, and precession.

Demonstration / Examples of Conservation of Angular Momentum

(a) Example is a man sitting on rotating stool. Initially he, stretches his arms far wide and then gives himself a push to rotate with the stool. Slowly he retracts his arms and people observe that his rotational velocity has increased without any external help!

The physics behind this magic is:
————————————————
Initial Inertia (I) = Large (due to stretched arm)
Initial Angular Velocity (ω) = Low
————————————————
Final Inertia (I) = Low (arms are retracted)

So, Final Angular Velocity (ω) = High

(b) Gyroscope: Is a high mass rotor inside three rings (gimbals). Outer gimbals is connected to frame. Any change is direction of frame and the net Angular momentum changes. So the rotor adjusts itself to maintain the total angular momentum, as that initially.

Gyro-Compasses are used to measure the velocity of moving systems, like aeroplane and ships, where there is no fixed reference to measure velocity.