Effects of External Torques on the Angular Momentum of Asymmetric Bodies

[Filipeml]

TL;DR Summary

I'm exploring how the distribution of mass in an asymmetric body affects its angular momentum when external torques are applied off the center of mass. I'm particularly interested in understanding the relationship between these forces and the body's precession and orientation. Looking for insights on how to describe this interaction mathematically.

Hello.

I'm studying rotation in classical mechanics, and I have a question about how the mass distribution of an asymmetric body affects angular momentum when external forces, such as torques, are applied off the center of mass. I know that the moment of inertia depends on the mass and the axis of rotation, but I'm unclear on how these forces influence the precession and orientation of the body.

Can someone clarify this relationship and how to describe it mathematically?

Thank you!

 

[jbriggs444]

I'm studying rotation in classical mechanics, and I have a question about how the mass distribution of an asymmetric body affects angular momentum when external forces, such as torques, are applied off the center of mass.

The mass distribution is irrelevant to the effect that an external torque has on the angular momentum of a body. The rate of change of angular momentum is equal to the external torque. It does not matter whether the body is symmetric or asymmetric. This relationship remains.

Orientation is trickier. The orientation of a body can change in unexpected ways, even in the absence of an external torque. See https://en.wikipedia.org/wiki/Precession#Torque-free_or_Torque_neglected

Non-rigid objects such as falling felines can change orientation with surprising speed. This without angular momentum or any external torque.

 

[Filipeml]

Thank you for your response and the helpful clarification! I appreciate you pointing out that the mass distribution doesn’t affect how external torque influences angular momentum—it really helps to simplify the concept.

I checked out the link on torque-free precession, and it’s fascinating how orientation can change even in the absence of external torque. This brings up another point I'm curious about: in the case of asymmetric bodies, if the torque-free precession is happening, does the mass distribution affect the rate or pattern of that precession? Or is it still completely independent of the body’s asymmetry?

Also, the mention of non-rigid objects, like falling felines, made me think: do the principles of rigid body rotation still apply in a simplified way to complex systems like these, or is there a separate set of rules for objects that can deform or change shape?

Thanks again for your insight! Looking forward to hearing your thoughts

 

[jbriggs444]

Thanks again for your insight! Looking forward to hearing your thoughts

In the case of a non-rigid object it is not clear that there is even a meaningful notion of "orientation".

In the case of a rigid object, the precession rate is given by formulas in the article that I referenced earlier. The mass distribution is reflected in the moment(s) of inertia. The normal "moment of inertia" that one encounters in first year physics is calculated in two dimensions about a chosen reference axis. The result is a single number. In a full three dimensional treatment one has a moment of inertia tensor calculated in three dimensions about a reference point. In general, a "tensor" is an array of numbers. In the case of the moment of inertia tensor, it is a three by three matrix.

I have not had any formal training in tensors. From what I have seen, matrix math figures heavily into their use. For instance, one can express a rotation as a vector (a three by one matrix). Multiply by a tensor (a three by three matrix) and out pops an angular momentum as a vector (a three by one matrix).

Note that a mathematician would likely cringe at the claim that tensors are arrays of numbers. Similarly, they might cringe at a claim that vectors are ordered tuples of numbers. The array of numbers in a tensor is merely a representation of the tensor in a particular coordinate system. Similarly, the tuple of numbers in a vector is merely a representation of the vector relative to a coordinate basis. But those concerns can perhaps wait for a course in linear algebra.

Again, I am untrained in tensors and have a limited knowledge of linear algebra.