Before we start Investigation 10, a few
concepts need to be explained.
Angular Velocity, 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' are added to give an overall
value. Obviously, 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 that if you
let go of the bar, you cannot start to 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 Iw2 where w 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.
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). 
Conservation
of Angular Momentum