# How many different axes of rotation can coexist?

There is a sphere in space. I can apply a force to cause the sphere to rotate around a central axis. An infinite number of possible central axes can be drawn.

1. Can I apply a force and then another force such that the sphere will rotate around 2 different central axes at the same time? I think yes.

2. Is there an upper limit to how many different axes of rotation a sphere can have at the same time? Or do various axes (all axes?) somehow cancel out or add up, like linear vector addition - even though 3 different forces contributed to my linear motion the net effect on me can be expressed by a single vector.

3. If 1 is true, and there are no external influences (whatever force got sphere rotating has stopped) will motion of the sphere change such that rotation is just around just one axis over time?

• Rotations are about a single axis. This axis may not be aligned about any one of your basis vectors but it's a single axis nevertheless. Mar 29, 2017 at 14:39
• A rigid body has six degrees of freedom, three translational and three rotational. The three rotational can be combined to a single vector, giving your axis. See the Wikipedia article on Rotation formalism in three dimensions. Mar 29, 2017 at 14:54
• A single force will make the body rotate about a point away from the center of mass. Only a pure couple causes rotation about the CM. Mar 30, 2017 at 13:53
• 1. For Euler's rotation theorem, see physics.stackexchange.com/q/19201/2451 and links therein. 2. 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/31475/2451 and links therein. Oct 13, 2017 at 16:33
• The sphere itself can only rotate around one axis at a time. But the axis of rotation itself can also rotate around an axis. This is called "precession". Sep 18, 2019 at 20:25

Can I apply a force and then another force such that the sphere will rotate around 2 different central axes at the same time?

No, this is not the case. Any rigid body, at any time, can only be rotating about one instantaneous axis of rotation. If you apply additional torques this axis can shift, but there's no such thing as having more than one axis of rotation.

Now, that said, if the body is asymmetric, like, say, a slab of wood, then you can think about spinning it quickly about its long axis and then more slowly about an axis orthogonal to that, but even then that's an illusion: at any given time, the block is undergoing an instantaneous rotation about a single axis, with the funky property that this axis will shift position with respect to both the body and the inertial laboratory frame.

In general, the rotational motion of the body is described by the direction $\hat{\mathbf n}$ of this axis and the angular velocity $\omega$ of the rotation, which get combined into a single vector $\boldsymbol{\omega}=\omega \hat{\mathbf n}$ for convenience. In the absence of torques, this angular velocity vector is not conserved; instead, the body rotates with constant angular momentum $$\mathbf L=I\boldsymbol\omega,$$ where $I$ is the moment of inertia matrix for the body; the rotational motion also conserves the rotational kinetic energy $E=\frac12 \boldsymbol{\omega}\cdot \mathbf L=\frac12 \boldsymbol{\omega}\cdot I \boldsymbol{\omega}$. That's about all that you can say in the general case, though if you move to a body-fixed frame you can analyze the motion a bit more understandably: there the angular momentum moves about (because the frame is not inertial) but it conserves both the energy and the total angular momentum $L^2$, which confines it to well-defined curves as described previously here and here on this site.

For the specific case of a sphere, then yes - when free of torques, both $\mathbf L$ and $\boldsymbol\omega$ will stay constant and the sphere will rotate with constant angular velocity about a fixed axis.

1. No. If you try to rotate a sphere already rotating around a central axis (spinning) by applying another (tangential) force, it will assume a new central axis of spinning. A body cannot spin around two axes at the same time.

2. There is no upper limit on the number of rotational axes apart from the number of spatial dimensions. For a simplistic example, a body spinning about the x-axis can revolve around a non-central y-axis and this whole system can revolve around another non-central z-axis. An infinite number of these combinations are possible.

3. See 1.

• It seems to me that 1 and 2 contradict each other. If a "body spinning about the x-axis can revolve around the y-axis" is that not spinning around 2 axes at the same time? Mar 29, 2017 at 15:53
• Edited. The y and z axes of rotation of the body (from the example) spinning around a central x-axis would not pass through it's centre. So the spinning body would be rotating eccentrically around the other axes. Mar 29, 2017 at 16:02

In the case of sphere ,you say, I think forces will just add up vectorialy and single axis of rotation will be achieved