Instability of free rigid body rotation about middle axis

[Jack Davies]

Hi everyone, I was recently talking to someone with a non-maths background about rotational stability, in particular how rotation is stable around the largest and smallest principal moments but not the intermediate one. He asked me if there was any 'obvious' reason for this, but one didn't spring to mind.

Obviously to anyone with a mathematical background you would tell them to write down Euler's equations and linearise them for a small perturbation. But I was curious if someone knew of any physical reason which would enable explanation to someone without a maths background.

 

[A.T.]

Obviously to anyone with a mathematical background you would tell them to write down Euler's equations and linearise them for a small perturbation. But I was curious if someone knew of any physical reason which would enable explanation to someone without a maths background.

Can't one explain the small perturbation without math, using a simple body, like two masses connected with a rod.

 

[Meir Achuz]

That is why we use math.

 

[FactChecker]

Can't one explain the small perturbation without math, using a simple body, like two masses connected with a rod.

I tried thinking along those lines, but I was getting the wrong intuitive answer. It is treacherous.

Here is an interesting approach that is more geometric (starting at ~ 1:15) than algebraic. But there is still math graphing involved.
NOTE: This youtube video is a little careless. Both surfaces are ellipsoids. The intersections of the surfaces retain the significant characteristics that are needed to make the main point.. You can visualize the axes being scaled so that one of the ellipsoids is a sphere.

 

[A.T.]

I tried thinking along those lines, but I was getting the wrong intuitive answer. It is treacherous.

Here is an interesting approach that is more geometric (starting at ~ 1:15) than algebraic. But there is still math graphing involved.

Here is more visualization of the geometric approach: