Understanding Angular Momentum in Rigid Body Rotation

[mr.physics]

Why can only the angular momentum of a rigid body rotating about an axis of symmetry be expressed as the product of the body's moment of inertia and its angular velocity?

 

[Meir Achuz]

The "moment of inertia" is actually a tensor. If a body is rotated about an axis that is not a symmetry axis (or, more generally, an axis for which the tensor is not diagonal), the angular momentum in not in the direction of the axis of rotation.
This is discussed in Mechanics terxtbooks.

 

[mr.physics]

Hmm.
I still don't really understand why not and my textbook doesn't supply much of an explanation.
Could someone please explain more thoroughly in terms a high school physics student might be privy to?

 

[cesiumfrog]

If a body is rotated about an axis that is not a symmetry axis [..] the angular momentum [is] not in the direction of the axis of rotation.

Would you describe (or cite) an example of that?

 

[PJC01]

Angular momentum is a fundamental concept in physics that describes the rotational motion of a rigid body. It is defined as the product of the body's moment of inertia and its angular velocity. This relationship holds true for a rigid body rotating about an axis of symmetry because of the nature of rigid body rotation.

A rigid body is one that maintains its shape and size while undergoing rotation. This means that all points on the body move in a circular path with the same angular velocity. In other words, the angular velocity of a rigid body is constant and does not change with time.

Now, the moment of inertia is a measure of a body's resistance to rotational motion. It depends on the mass distribution of the body and the distance of the mass from the axis of rotation. For a rigid body rotating about an axis of symmetry, the mass distribution is symmetric and the distance of the mass from the axis of rotation is constant. Therefore, the moment of inertia remains constant throughout the rotation.

Based on these two characteristics of rigid body rotation, we can see that the product of the moment of inertia and the angular velocity remains constant as well. This is because any change in the angular velocity would require a change in the moment of inertia, which is not possible for a rigid body. Hence, the angular momentum of a rigid body rotating about an axis of symmetry can only be expressed as the product of the moment of inertia and the angular velocity.

In summary, the reason why only the angular momentum of a rigid body rotating about an axis of symmetry can be expressed as the product of the moment of inertia and the angular velocity is due to the constant angular velocity and moment of inertia of a rigid body during rotation. This is a fundamental concept in understanding the dynamics of rigid body rotation and has important applications in various fields of science and engineering.