# Two axes for rotational motion

I understand that angular momentum is a vector, etc..

But, what really happens when some object, say a ball for example, is set to rotate along two axes? What would the resulting motion look like?

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For the Dzhanibekov effect, the tennis racquet theorem, and the intermediate axis theorem, see e.g. physics.stackexchange.com/q/17504/2451, physics.stackexchange.com/q/34364/2451, physics.stackexchange.com/q/67957/2451, and links therein. – Qmechanic Dec 5 '12 at 19:45

It looks just like rotation around a different axis with a different rotational speed. Specifically, if you set an object to rotate with angular velocity $\vec\omega_1$ and also with angular velocity $\vec\omega_2$, then it's really rotating with angular velocity $\vec\omega_1 + \vec\omega_2$. The direction of the vector $\vec\omega_1 + \vec\omega_2$ is the overall axis of rotation of the object.

Euler's rotation theorem guarantees that any rotation of a rigid object can be expressed as a rotation around a single axis.

All of this applies instantaneously, in the sense that at any given moment, the body is rotating around a single axis. It is possible that the direction of the rotational axis changes over time, though, and this can lead to more complicated motions that may seem as though they can't be described by single-axis rotation.

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This can be misleading. See math.stackexchange.com/q/44696 – leonbloy Jan 6 '12 at 23:53
I'll grant that it's not always obvious how to apply this correctly. But note that in the example of the cylinder in that question, one of the rotational velocities is changing its direction over time. That's why you can't describe the overall motion as rotation around a single, fixed axis. – David Z Jan 7 '12 at 0:13
Agree. But it happens often IMO, when someone speaks of an object that has "two superposed rotations", that we have two rotation axes that are fixed with respect to the body (but not to the reference space!), it's important to note that in that case you answer does not apply. – leonbloy Jan 7 '12 at 12:42
No, everything I've said still applies instantaneously whether or not the rotation axis is fixed with respect to the reference space. I suppose it's worth clarifying that, though. The rotating cylinder example is actually more complex because you have rotation about one axis that is fixed with respect to the body and one that is fixed with respect to the reference space. – David Z Jan 8 '12 at 0:06

Angular rotation is a vector so at any given instant any rigid body can only be rotating about one axis. If the body is rotating freely in space with no external forces then angular momentum is conserved. If the object is spherically symmetrical like the ball you suggest as an example, then the angular velocity is in the same direction as the angular momentum and its motion can only be a simple constant rotation about one axis.

For a more complex asymmetric rigid object the moment of inertia is a symmetric matrix with three perpendicular principle axis. If the rotation aligns with one of these axis it will still have a constant angular velocity, but if not then the angular velocity can itself change direction even while the angular momentum remains constant. There are cases where the angular velocity vector processes around the direction of the angular momentum. This makes it look like it has more than one axis of rotation but really it is one axis that is itself rotating. Here is an animation video to show this

More complex motion is possible when all three axis are different as seen in this animation

In this last video a book which has three different principle moments of inertia is used on the space station to demonstrate some of the variety of motion possible.

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– voix Jan 7 '12 at 7:47
Thanks, I was looking for that one but it does not have good keywords. For this to work the body must have three different principle moments and you set it rotating about the axis for the middle of the three. – Philip Gibbs - inactive Jan 7 '12 at 8:46

A rigid body can only rotate by one axis and stay rigid. In fact, the only allowed motion is a screw, whereas a rotation about an axis happens simultaneously as a translation along the same axis (called a twist). Their relationship is called the screw pitch. A pure rotation has pitch=0.

Now if you are asking what if you have a joint that allows two or more rotations (like a universal joint) then the results is that at any instant, there is only one effective rotation axis.

If you have a sequence of three rotations, with rotation matrices $R_1$, $R_2$ and $R_3$ each one about a local axis $\hat{z}_1$, $\hat{z}_2$ and $\hat{z}_3$ the the total angular velocity vector is

$$\vec{\omega} = \hat{z}_1 \dot{\theta}_1 + R_1 \left(\hat{z}_2 \dot{\theta}_2 + R_2 \left( \hat{z}_3 \dot{\theta}_3 \right) \right )$$

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Rotation is geometrically possible only over one axis. This axis can change in time, but at each instant it will be one.

This is geometrical property of 3d space.

Angular momentum axis does not coincide with rotation axis. Generally, rotation axis do precess around angular momentum axis.

Here is the example of rotating body whose angular momentum is absolutely constant, but rotation axis does vary: