# Torque Free Rotation again

1. Apr 21, 2014

### fog37

Torque Free Rotation....again....

Hello Forum,

I am still unclear on how, from the inertial (lab) frame of reference, the moment of inertia I, which is a tensor with 9 components, changes with time t.....

A rigid body moving in the air changes its coordinates relative to the origin of a fixed lab frame of reference. Why would the moment of inertia change too?

Torque free means zero net torque acting on the rigid body. But torque is a quantity that depends on the choice of the pole about which torque is calculate: from the lab frame, the choice of different poles will lead to different numerical values for the torque, correct?

It is possible to diagonalize the moment of inertia and find the 3 principal directions. If the moment of inertia has components that are not constants but instead depend on position (x,y,z), we will find a different triad of principal axes for each different point P since there is a different inertia tensor for each different point P, correct?

thanks,
fog73

2. Apr 21, 2014

### Simon Bridge

Welcome to PF;
It's related to this:

As for the math details - why not try it and see?
Pick an example.

I have a bunch of notes somewhere ... oh here it is:
http://home.comcast.net/~szemengtan/ClassicalMechanics/SystemsAndRigidBodies.pdf [Broken]

Last edited by a moderator: May 6, 2017
3. Apr 28, 2014

### fog37

Thanks Simon.

I guess from the fixed lab frame of reference, the moment of inertia does change while it does not change from body frame of reference.

From the lab reference frame angular momentum is constant while the angular velocity ω precesses and the moment of inertia I changes with time....

Thanks!

Fog37

4. Apr 28, 2014

### Simon Bridge

Note: The body frame is not inertial.

Being careful to be specific about frames is the way to make progress here - you'll see Sze Tan does this in the lecture notes.