If two stones are connected with a string and one of them is hurled, the other stone will fly after the first at the end of the stretched string. Each stone will pass the other, and this forward motion will be accompanied by a rotation. Let us forget about the gravitational field—assume that the throw was made in interstellar space.

The forces acting on the stones are equal in magnitude and directed towards each other along the string (for these are forces of action and reaction). But then the lever arms of both forces with respect to an arbitrary point will also be the same. Equal lever arms and equal but oppositely directed forces yield torques which are equal in magnitude and opposite in sign.

The resultant torque will be equal to zero. But it follows from this that the change in angular momentum will also equal zero, i.e. that the angular momentum of such a system remains constant.

We only needed the string connecting the stones for visualization. The law of conservation of angular momentum is valid for any pair of interacting bodies, no matter what the nature of this interaction.

Yes, and not only for a pair. If a closed system of bodies is being investigated, the forces acting between the bodies can always be divided up into an equal number of forces of action and reaction whose moments will cancel each other in pairs.

The law of conservation of total angular momentum is universal, it is valid for any closed system of bodies.

If a body is rotating about an axis, its angular momentum is \[N = m \, vr\] where \(m\) is the mass, \(v\) is the speed, and \(r\) is the distance from the axis. Expressing the speed in terms of the number \(n\) of revolutions per second, we have: \[v = 2 \pi nr \quad \textrm{and} \quad N = 2 \pi mnr^{2}\] i.e. the angular momentum is proportional to the square of the distance from the axis.

Sit down on a swivel stool. Pick up heavy weights, spread your arms wide apart and ask somebody to get you rotating slowly. Now press your arms to your chest rapidly-you will suddenly begin rotating faster. Arms out-the motion slows down, arms in-the motion speeds up. Until the stool stops turning because of friction, you will have time to change your rotational velocity several times.

Why does this happen?

For a constant number of revolutions per second, the angular momentum would decrease in case the weights approached the axis. In order to “compensate” for this decrease, the rotational velocity increases.

Acrobats make good use of the law of conservation of angular momentum. How does an acrobat turn a somersault in mid-air? First of all, by pushing off from an elastic floor or his partner’s hand. When pushing off, his body bends forward and his weight, together with the force of the push, creates an instantaneous torque. The force of the push creates a forward motion, and the torque causes a rotation. However, this rotation is slow, incapable of impressing the audience. The acrobat bends his knees. “Gathering his body” closer to the axis of rotation, the acrobat greatly increases the rotational velocity and quickly turns over. This is the mechanics of the somersault.

The movements of a ballerina performing a succession of rapid turns are based on this same principle. Ordinarily the initial angular momentum is imparted to the ballerina by her partner. At this instant the dancer’s body is bent; a slow rotation begins, then a graceful and rapid movement-the ballerina straightens up. Now all points of her body are closer to the axis of rotation, and conservation of angular momentum leads to a sharp increase in speed.