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It appears that this is kinematically possible, but it looks quite unprobable. For example, it is quite improbable for a sphere to have rotation about three mutually perpendicular axes (supposedly passing through its centre) simultaneously.

This led me to think if this is even possible kinematically.

As a simple case, consider a particle attached to a three dimensional reference frame. Now, It is possible for the particle to move in all three axes simultanoeusly. Does this necessarily imply that the particle can have rotation about all three axes simultaneously?

Shubham
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    Yes. – ACuriousMind Oct 05 '14 at 22:04
  • Yes, but how do I know? – Shubham Oct 05 '14 at 22:10
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    Take a rigid body into your hand, rotate it around one of its three axes. Can you replace this rotation for a fully asymmetric body with combinations of the other two? – CuriousOne Oct 05 '14 at 22:13
  • No, but how does this imply that 3 independent rotations are possible? – Shubham Oct 05 '14 at 22:20
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    Possible duplicates: http://physics.stackexchange.com/q/19201/2451 and links therein. – Qmechanic Oct 05 '14 at 22:20
  • You may be confusing two notions. Any combination of rotations may (over an infinitesimal period) be replaced with a single rotation about an appropriately chosen axis, but that does not mean that the general non-infinitesimal case can be expressed in that way. – dmckee --- ex-moderator kitten Oct 05 '14 at 22:57
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    @dmckee - Of course they can. Rotations in three dimensional space are described by the rotation group SO(3). – David Hammen Oct 05 '14 at 23:17
  • @DavidHammen I think perhaps you have misunderstood me---or I have not written what I thought I did. Any infinitesimal rotation can be described as a single infinitesimal rotation around a single axis (which takes 3 parameters, but can be conceived of as a "single" axis of rotation). A general finite rotation can also be described by three parameters, but can not be described by "we rotate by this amount around this single axis" the way the infinitesimal case can. – dmckee --- ex-moderator kitten Oct 05 '14 at 23:25
  • @dmckee - It most certainly can. It's Euler's rotation theorem. – David Hammen Oct 05 '14 at 23:47
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    @dmckee - A simple demonstration. Imagine a sphere with which we have endowed an orthogonal set of axes, call them $\hat x$, $\hat y$, and $\hat z$. fixed with respect to the sphere. We can easily construct another set of axes, $\hat x'$, $\hat y'$, and $\hat z'$, such that the original $\hat x$ axis points in the direction of $\hat x' + \hat y' + \hat z'$. Now set the object rotating about the original $\hat x$ axis. Viola! In our new system, the object is simultaneously rotating about $\hat x'$, $\hat y'$, and $\hat z'$. – David Hammen Oct 06 '14 at 00:02
  • @DavidHammen You are right. I'm wrong. And I still haven't said what I meant to say. – dmckee --- ex-moderator kitten Oct 06 '14 at 00:37
  • @dmckee - If you are saying that there is no way to have everything but the center of rotation moving due to the rotation, you're right. And maybe that's what the OP is asking. You have to go to four dimensional space to see that kind of rotation (it's called a double rotation). – David Hammen Oct 06 '14 at 02:36

3 Answers3

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Made visualisations so you can see how it looks like.

Rotating X:

enter image description here

Rotating X and Y:

enter image description here

Rotating X,Y and Z:

enter image description here

Michal
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There is not a simple answer for this since it is both yes and no.

From an internal frame of reference:
Looking at the object, nothing is happening.
Looking at the outside world, it is moving in a predictable pattern.

From an external frame of reference:
The rotation on one axis is easy to see.
The rotation on 2 axes looks like a rotation on a moving axis.
The rotation on 3 axes is difficult to describe.

LDC3
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The answer is that at any moment a rigid body is rotating about an arbitrary axis, with an arbitrary angle. There are two components defining the direction of rotation axis, and one for the magnitude of rotation.

Often we transform these three quantities into three sequential rotations called Euler Angles. If you look up rotation matrix you will find all sorts of ways to go from axis angle angle to three angles. They both describe the same thing, but with different representation.

BTW, The pure motion of a rigid body is a simultaneous rotation about an axis and translation along the same axis. This is called a screw motion. See https://physics.stackexchange.com/a/86020/392 for more details.

Community
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John Alexiou
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