Imagine a rotating platform, this is rotating at w_{1} radians per
second. On this platform, and offset by 1m from the centre, is a smaller rotating
platform. This small platform is rotating at w_{2} radians per second
relative to the large platform.

What is the angular velocity of the small platform relative to the earth?

The answer is that it is w_{1} + w_{2} radians per second.

Note: this is different from the planetary motion example in that the whole frame of reference is rotating.

However this example is relatively simple because the axis of rotation is in
the same direction, so what happens if we rotate w_{2} by 45 degrees?

The smaller rotating platform now has a very complicated motion, I'm not sure
we can represent its rotation by an equivalent single angular velocity? This
example is made more complex by the translation which is also a function of
time, perhaps we can remove that effect by putting the smaller rotating platform
on the w_{1} axis as follows:

The smaller rotating platform still has a very complicated motion which does
not appear to be represented by a single angular velocity. Perhaps the best
way to analyze it is that the smaller rotating platform has a single angular
velocity at any instant in time (w_{1} + w_{2}), but this angular
velocity is continuously varying with time. So angular momentum is continuously
being changed in the system (so a closed system won't continue to rotate in
this complex way).

## Further Information

These pages contain examples to illustrate the principles involved, for an discussion of the theory about angular momentum see this page.