# Rotational momentum - anybody an expert?

Hi!

The reason I am asking the following few questions, is because I want to understand the way rotational momentum works in certain cases, so I can apply to activities I do in real life (snowboarding, diving).

My knowledge of this area extends as far as normal momentum, rotational momentum, moment of inertia, and centripetal force.

Take for instance, a backflip. Very easy to do. You gain rotational momentum backwards by pushing off the ground, and then you jump up while you have that momentum. Then you control the speed of spin by extending/contracting your body. Easy. Same goes for a frontflip.

180-1080 - normal sideways spins. Very similar to backflip. You throw your hands (or your shoulders for a slower spin) all in one direction, while your legs are pushing off the ground in the opposite direction. When you gain the momentum, you jump up, and the result is you spin. You control the speed of spin by stretching your arms/legs out and contracting them. You can throw counter-rotation when landing by rotating your torso in the opposite direction (this is very important in snowboarding - so you can land it just right).

Now I understand all that "simple" stuff. Now, as I've been advancing with snowboarding and diving, I am beginning to enter the "inverted spins" area, where you do things like bacfklip and 720 simultaneously. Now this is where I need help.

I guess the first question is - can you say that an object can have multiple axises of rotation? Like a vertical and a horizontal axis?

For instance - take a ruler. If you flip it and spin it at the same time - it seems that it has 2 axises. But if you take something less "ruler-like", such as a backpack, and try to do the same, it seems that you have only 1 messed up axis of rotation. But I guess this question is kind of dumb - it's like asking does an object that moves straight in diagonale have 2 velocities - forward and to the left, or does it have one - towards the corner. So I guess I answered my quesion myself.

The next question is, how can your "sideways" rotation affect your "upside down" rotation. For instance, I've tried doing a frontflip 720 off the diving board. IF I just combine a frontflip and 720 - throw my body forwards and spin it sideways at the same time - everything seems to be fine. But if I spin my hand around WHILE IN THE AIR, I only seem to be able to do a 360. Whereas if I spin it in some other direction, I end up doing some crazy sideways-flip 1080...

Like this is what's bugging me.

For instance, I was watching this skiing competition. Those guys do like a double backflip 720, which means for every backflip they do 360 rotation sideways. The wierdest thing is that when they take off the jump, the ONLY throw their body backwards - they don't put ANY sideways spin into the jump at all. And then when in the air they "STEER" with their hands. And the commentator says "watch her control those spins with her hands". It seems to me that they take of as if they were going for a normal double backflip, but then they somehow use rotation of their hands to add 720 into the jump, WHILE in the air. This seems to kind of defy the laws of physics - but the fact is they do it.

Another thing is a very popular "off-axis 720" spin, in both skiing and snowboarding. It's like they do a 720 spin, but their body is half way upside down... horizontal... when in the air. This also seems to defy laws of physics... It's like first they rotate 90 degress backwards and end up in "laying down" positon, while their body is spinning around, and then they rotate 90 degrees forward, back into standing position...

THis has been bugging me for quite a while, and I'd be very thankful if somebody could give me some insight on this aspect of physics, or forward me to a resource that would help me.

## Answers and Replies

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Ok let's observe

http://ablogic.net/files/inverted7.wmv [Broken]

Check out this video (if it opens as text file - run media player and go to File > Open from URL).

This guy does an off-axis 720. Watch what happens - he starts out PERFECTLY veritcal. It's like he's just standing up. Then half way through the jump, he rotates backwards 90 degrees, and looks like he's laying down in the air. Then as he lands, notice how he swings his hand, and his body comes out forward from "the side".

WHat the hell is the physics behind this?

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http://ablogic.net/files/rodeo.WMV [Broken]

check out this one. Also use media player to open it. Here this guy does a rodeo on mini skis. This one seems much simpler than the previous one - he simply throws the body upside down, and twists it as he takes off. Then to help his landing he throws a little counter rotation with his hand as he lands. This one I understand...

But the off axis 720 is ridicoulous. Does it defy fisics?

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arildno
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Hi, ivj:
Just one clarification:
In general, given a fixed coordinate system, an object may rotate about each of the 3 fixed axes simultaneously. However, you may always regard an object to have a single, instantaneous rotation axis (which changes through time).

pervect
Staff Emeritus
Science Advisor
Windows Media Player won't play your files for me :-(.

One thing you should remember is that angular velocities add like vectors. So it's possible to have a body with components of rotations around various axes. By using the usual vector formalism where the vector points along the axis, the result of the sum of the components will be a single rotation around some axis.

For torque-free motion, the angular momentum of a system will remian constant. The math gets complicated because as the body rotates, its primary axes change their orientation in space. The usual approach is to analyze the rotations in a body-centered frame via Euler's equations. And that math only describes rigid bodies, though it could in principle be extended to non-rigid bodies (but it would take more effort).

If you want to read about Euler's equations, you can try "Classical Mechanics" by Goldstein, but I'm not sure if your interest is that serious.

To make life interesting, some rotational modes or rigid bodies are stable, and some are not. If a body has three different moments of inertia about it's three primary axes, then rotation around the axis with the middle value of momeent will be unstable, rotation around the smallest moment axis will be marginally stable (it will be stable for a perfectly rigid body, but exhibit long-term instabilities if the body is not perfectly rigid), and finally rotation around the axis with the largest moment will be stable and represents the lowest energy state of the system for a given amount of angular momentum.

I do seem to recall there was some popular book, something like "the physics of sports" that at least talked about some of these issues. However, going to amazon.com, the book I find by that title doesn't seem to be the one I was remembering.