Conservation of Angular Momentum

If the external torque is responsible for a change in angular momentum, as we have just learned, then the absence of external torque must mean that the angular momentum of a system does not change. We have another conservation law in hand:

The total angular momentum is always conserved, if there is no net external torque acting on the system.

In this case we have:

$\Delta$L = L - L_o = I $\omega$ - I_o$\omega$_o = 0

I$\omega$ = I_o$\omega$_o

This is now the third major conservation law that we have encountered, after the conservation law of energy and the conservation law of linear momentum.

The angular velocity can change, but the moment of inertia must then change in such a way that the angular momentum remains a constant. This is used by ice-skaters to perform pirouettes, or by platform divers to control their rotation in the air.

Because angular momentum is a vector, conservation of angular momentum implies that the axis of rotation will also not change in the absence of external torques. This effect is the idea behind the use of gyroscopes to maintain absolute orientation in space.

If an external torque is applied to a spinning object, the direction of the axis will change. This change is called precession. The spinning Earth, for example, has an angular momentum vector which precesses with a period of 26,000 years.

Remark:

Precession is outside the scope of the topics covered in this class, and the videos included here are only information items.