Why Does the Moment of Inertia Change in Torque-Free Rotation?

[fog37]

Torque Free Rotation...again...

Hello Forum,

I have read an old, but good, thread about torque free rotation:

https://www.physicsforums.com/showthread.php?t=405781

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

 

[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

 

[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

 

[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.

 

[PJC01]

Hello fog73,

Thank you for bringing up this interesting topic. You are correct in saying that the moment of inertia is a tensor with 9 components and that it can change with time. This is because the moment of inertia depends not only on the mass distribution of the rigid body, but also on its orientation and position in space.

In the case of torque-free rotation, the net torque acting on the body is zero, but this does not mean that there are no individual torques acting on different points of the body. As you mentioned, the choice of pole about which torque is calculated can affect the numerical value of the torque. However, the net torque will always be zero as long as the body is rotating without any external forces.

Regarding your question about the moment of inertia changing with position, this is also true. As the body moves in space, its position and orientation change, leading to a change in the moment of inertia. This is why it is important to consider the moment of inertia as a function of both time and position.

I hope this helps clarify the concept of torque-free rotation and the role of the moment of inertia. Keep exploring and asking questions! Science is all about curiosity and seeking understanding.