# Rotation of rigid body - wobbling

1. May 7, 2008

### Nick R

When a rigid body experiences a rotation about an axis other than one of its "principal axis of rotation", it experiences a "wobble"

I have been trying to understand why this is so (intuitively).

Here is what I have come up with - please tell me if I have misconceptions or misunderstandings, or if I'm just full of crap :(

I have come up with this qualitative explanation - the particles that comprise the rigid body are subject to two constraints - they must rotate about $$\omega$$, but at the same time they must stay fixed relative to each other. The latter constraint introduces the wobble when the rotation is not about a "principal axis of rotation".

Is that a correct way of looking at it?

A more mathematical explanation I have come up with is (tell me if any of these concepts are flawed... hopefully this makes some sense without a drawing) -

Rigid body experiences a rotation about $$\omega$$, where $$\omega$$ is not parallel to a principal axis of rotation.

Viewing the system from a coordinate system that is fixed with respect to the "inertial tensor" I of the rigid body,
$$\omega$$ actually appears to rotate about a principle axis

(imagine instead rotating the coordinate system about $$\omega$$ - $$\omega$$ effectivly rotates about a principal axis(s)).

Therefore, because by definition

L = I $$\bullet$$ $$\omega$$

coordinate system is fixed with respect to inertial mass tensor I
so dI/dt = 0

but $$\omega$$ is not fixed in the coordinate system so
d$$\omega$$/dt $$\neq$$ 0

therefore, dL/dt = torque $$\neq$$ 0

And this torque is responsible for the wobble.