Consider a particle of the body of mass m

Therefore the angular momentum of m

If we add together the angular momentum for all the particles of the body then:

angular momentum of the body = Σmωr

The units for angular momentum are kg
m^{2} radian s^{-1}

You should have expected a result like this, as it
is similar to that for linear momentum (mu). (I taking the place of m and ω taking the place of u)

In the same way that if a force is
applied to a body for a certain time it will change the linear momentum of a body the
application of a couple C for a certain time t will change the angular velocity from ω_{o} to ω_{1} and so give a change of angular
momentum of the body such that:

A couple of moment 7.5 Nm is applied for 5s to a disc of moment of inertia 0.15 kgm

Ct = IΔ

7.5x5 = 0.15 x Δω

Change in angular velocity (Δω) = 250 rad s

A useful term when considering
rotating objects is one known as the radius of gyration. We can write the moment of inertia I
of a body as Mk^{2}_{1} where M is the mass of the body and k is a distance
called the radius of gyration of the body. For example, for a disc the moment of inertia is
Mr^{2}/2, and so k = (r^{2}/2)^{1/2} = r/2^{1/2} .

(see
Figure 2).

The radius of gyration can be defined as the distance from the centre of
rotation of a rotating body to the point where the mass can be considered to be concentrated
(not to be confused with the centre of mass).

The angular momentum of an isolated system is constant.

This may be expressed in a formula as:

Iω

where Iω

A very simple demonstration of this law can be shown by a person standing on a rotating
platform. If they first rotate with their arms outstretched and then bring their arms in, their
angular velocity will increase. (Figure 3).

This happens because their angular
momentum must remain constant, the decrease in their moment of inertia as they drop their
arms results in an increase in their angular velocity. The effects are much more impressive if
the person holds a heavy book or a 1 kg mass in each
hand.

Many of you will have experienced the effects of the
conservation of angular momentum when riding a bike. It is very hard to keep the bike
upright when it is stationary, but as soon as the bike is moving and its wheels are rotating it
becomes much easier.

The rotation of the wheels produces angular momentum and prevents
the bike from falling sideways, because this would give a change in the angular momentum
(since this is a vector quantity).

A helicopter is prevented from rotating about a vertical axis either by the small tail rotor or by a sideways jet of air ducted through the tail
boom.