Conservation of Angular Momentum, with Examples

Section 5.6 established that the total angular momentum of any system of particles, about a fixed

point or a fixed axis, changes only under the action of a net EXTERNAL torque:

The law of conservation of angular momentum follows immediately as the special case $\vec\tau_{ext} = 0$: whenever the net external torque acting on a system, about a chosen point or axis, is zero, the total

angular momentum $\vec L$ of the system about that same point or axis stays exactly constant -- however

the system's own internal distribution of mass, and therefore its own moment of inertia, may change over

time.

This last clause is the key to every everyday example of the law. Since $L = I\omega$ for rotation about a

fixed axis, conservation of $L$ directly means

for any two instants $1$ and $2$ at which no external torque has acted in between -- so if a spinning

system's OWN moment of inertia $I$ changes (by the system itself pulling its mass closer to, or pushing it

farther from, the axis), its angular velocity $\omega$ must change too, in exactly the opposite sense, to

keep the product $I\omega$ fixed.

The spinning skater or the stool-and-arms example (see the accompanying figure) is the standard

illustration: a person spinning with arms stretched far out has a relatively large moment of inertia

$I_1$ and spins at some angular speed $\omega_1$; pulling the arms in close to the body redistributes the

same total mass much closer to the rotation axis, sharply reducing the moment of inertia to some smaller

value $I_2 < I_1$. Since no external torque acts about the (frictionless, vertical) spin axis during this

purely internal rearrangement, $I_1\omega_1 = I_2\omega_2$ must still hold, so the angular speed must

correspondingly INCREASE to $\omega_2 = I_1\omega_1/I_2 > \omega_1$ -- exactly the familiar sudden speeding

up seen the instant the arms are pulled in, and the equally familiar slowing down the instant they are

flung back out.

The identical principle governs a diver curling into a tight tuck to spin faster mid-air and then

straightening out again to slow the spin before entering the water, and a spinning ballet dancer or figure …