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Imagine an object floating in space with no significant gravitational forces. We poke the object a few times from various angles. It will end up spinning in a certain way.

Is it the case that this "spin" will always be a constant angular velocity about a particular axis?

My gut feeling is that the answer is yes, especially due to Euler's rotation theorem.

I tried to prove it by first simplifying the object to a sphere. Then saying that a particular point on the sphere must follow a path that never crosses itself -- because the "next" and "previous" points are always unique, a crossing would indicate that a certain point has 2 potential "next" points. Therefore you have a bunch of parallel paths which run parallel to an axis. This doesn't really feel like a solid proof though. (What if there is a wobble that results in a series of wobbly lines that don't cross.)

Is it correct to say that every "poke" to the object applies a torque, and the torque vectors can simply be added up to result in the final angular velocity?

CaptainCodeman
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2 Answers2

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Well -- no.

Here's a particularly good example on YouTube. It shows a T-handle being spun along the intermediate axis of its moment of inertia, and then exhibiting spin instability by flipping the direction it's pointing, all while continuing to spin.

The preserved quantities in this case are energy and angular momentum (with respect to the inertial frame of reference). But the spin referenced to any axis that you pin onto the object doesn't have to be constant.

But you were close -- if you add up the rotational impulses that you impose on the object (basically the torque integrated over time), the result will be the angular momentum.

TimWescott
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  • If the rotational impulses add up to the angular momentum, when and how does it decide to convert angular momentum to another type of energy and vice versa? – CaptainCodeman Dec 26 '22 at 02:51
  • Momentum isn't energy, and energy isn't momentum. If the object isn't coupled to anything else, then neither energy nor momentum can be transferred. – TimWescott Dec 26 '22 at 04:35
  • Isn't there some sort of kinetic energy associated with momentum though? – CaptainCodeman Dec 28 '22 at 04:00
  • Kinetic energy is a result of motion (that's where 'kinetic' comes from), and momentum is a result of motion. But they are two very different things. This is basic physics stuff, that should show up in a high-school AP physics class, or a college introductory physics class. – TimWescott Dec 28 '22 at 15:52
  • it doesn't matter if it's basic or not, there is no need to condescend. – CaptainCodeman Dec 29 '22 at 14:32
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No. In general, if the resulting angular momentum vector doesn't coincide with one of the body's principal axes, the rotation will be wobbly, perhaps extremely so.

A sphere has a degenerate moment of inertia tensor: inertia is the same around any axis. In this special case, the rotation won't wobble.

Note that applied physicists call the wobble "nutation" while theoretical physicists call it "precession".

John Doty
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