Introduction to Rotating Systems

Before we start Investigation 10, a few concepts need to be explained.

Angular Momentum and Moments of Inertia

This is similar to the concept of linear momentum and inertia. Inertia is a measure of how difficult something is to get moving or to stop. By the same definition, the Moment of Inertia is a measure of how difficult something is to set into rotational motion, or to stop that motion. It is a scalar value calculated by the mass of the particle, multiplied by the square of its distance from the axis of rotation. In a person, the contributions of the Moment of Inertia of all the 'particles' which make them up are added to give an overall value. A heavier, more extended body has a larger Moment of Inertia than a light compact one.

Angular Velocity is a vector quantity, represented by an arrow whose length is the rotational speed and the direction is parallel to the axis of rotation. The direction the arrow points in depends on the direction of rotation, in accordance with the right hand screw rule (if the person was a screw, rotating as if being tightened, then the Angular Velocity arrow points in the direction that the pointed end of the screw points).

Angular Momentum is given by the product of the Angular Velocity about a given axis, multiplied by the moment of inertia about the same axis (it is therefore a vector quantity, in the same direction as the Angular Velocity vector.

Somersaulting and Twisting

Somersaulting is where the body rotates head over heels, through an axis through the waist (if body is straight, this is where centre of gravity is).

Twisting involves spinning or pirouetting, rotating about an axis from head to toe.

If a performer is both somersaulting and twisting at the same time, then he has an associated angular momentum for each. Since both are vector quantities, his total angular momentum is found by adding the two vectors.

Conservation of Angular Momentum

Newton's third law can be applied to a rotating system to state that:

'In the absence of external impulses, the total angular momentum in a system is conserved.'

This is a problem for trapeze artists (and divers and gymnasts etc.). What it means is that once you let go of the bar, you cannot start a twist or somersault in the air. Since you will have had not angular momentum to begin with, you cannot 'create any' in the air, since there are no impulses on your body (air resistance may be considered negligible).

It should be stressed that conservation of angular momentum does NOT imply conservation of angular velocity. Since momentum is given by Iω² (where ω is the angular velocity), by changing I, you can increase or decrease your speed of rotation.

This principle applies to the trapeze, but is best demonstrated with a more classical example of an ice-skater. A skater pirouetting on the ice starts with her arms straight out, this gives her a large I (moment of inertia) and once she sets herself spinning, this gives her a large angular momentum, but a low speed. She then pulls her arms in, reducing I. Since angular momentum is conserved, her angular velocity must increase to compensate, so she spins faster, achieving the pirouette.