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I am a bit new to the concept of the Inertia Tensor. One question that comes to mind is, what is the physical meaning of the off-diagonals i.e. the products of inertia?

I read this post : https://www.physicsforums.com/threads/physical-meaning-of-product-of-inertia.401927/post-2711721

and I find it quite helpful.

This seems to connect with the idea of dynamic balancing, i.e. an object is dynamically balanced IFF the rotation axis lines up with a set of axes for which the inertia tensor is diagonal. (Have I got that right?)

But I'd like to see a toy problem that exemplifies this in a simple way, and in particular, I'd like to know how it ties in with the following formula:

which is from Wikipedia https://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_tensor

If there are some ##m_k x_k z_k## terms that don't all cancel out, how does that call for an external torque to maintain the rotation axis?

I read this post : https://www.physicsforums.com/threads/physical-meaning-of-product-of-inertia.401927/post-2711721

and I find it quite helpful.

Products of inertia certainly do have physical meaning. Suppose the Ixy for some object is non-zero. That means that the object cannot perform a pure x-axis rotation without some external torque. (The same also pertains to a pure y-axis rotation.)

This seems to connect with the idea of dynamic balancing, i.e. an object is dynamically balanced IFF the rotation axis lines up with a set of axes for which the inertia tensor is diagonal. (Have I got that right?)

But I'd like to see a toy problem that exemplifies this in a simple way, and in particular, I'd like to know how it ties in with the following formula:

which is from Wikipedia https://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_tensor

If there are some ##m_k x_k z_k## terms that don't all cancel out, how does that call for an external torque to maintain the rotation axis?

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