a question about rotating bodies

[technetium]

Hi Forum,

 

I hope you can help settle a bet. The question is:

 

Can a body (say a ball) spin about multiple axes at the same time? For example imagine a ball with its centre at the origin of x-y-z coordinates. If the ball is rotating about the X axes is it possible for it to be simultaneously rotating about the y and z axes as well?

 

If it is possible is the resultant rotation simply the "vector" sum (or equivalent) of each of the individual rotations, just as the velocity of a moving body can be resolved into x, y & z components, and if so how does one "add" the individual components (direction and "speed" of rotation) to get a resultant spin?

 

Thanks

[D H]

Can a body (say a ball) spin about multiple axes at the same time?

Yes, no, and yes.

 

The first yes answer: The x, y, and z labels we attach are not real; they are inventions of our minds to help us explain what is going on. Defining a set of orthogonal axes does not magically constrain a rigid body to rotate around those axes and those axes only. Suppose you come up with a coordinate system. Suppose your friend doesn't like your coordinates; he likes some other set -- say your coordinates rotated by 45 degrees about your z axis. If you give an object a rotation around your x-axis, your friend will see the object rotating about an axis that is not purely x or y in his coordinate system.

 

The no answer: The above is just nomenclature, however. At any point in time, the motion of a rigid body can always be described in terms of the translational velocity of the body's center of mass plus a rotation about an axis passing through the body's center of mass. Note the word always. At any point of time, the rotation of a rigid body can always be described as being about a single axis. This is called the eigen axis, or axis of rotation. Your friend, for example, sees the object rotating about the axis [math]\hat{\boldsymbol{\omega}} = 1/{\sqrt 2} (-\hat {\boldsymbol x} + \hat {\boldsymbol y})[/math]. Does this mean the object is simultaneously rotating about his x and y axes? That is a question of semantics, not physics. The important point is that the body is rotating about a single axis.

 

The second yes answer: That axis of rotation can change. Think of a precessing top. The axis of rotation is itself rotating. Does this mean the top is rotating about multiple axes? Again, this is a question of semantics, not physics.

 

 

Bottom line: Your bet is a question of semantics, not physics. The important thing to note is that at any point in time the motion of a rigid body can be described in terms of rotation about a single axis. This is not the case for a non-rigid body -- a string of spaghetti, for instance.

[granpa]

in four dimensions it can rotate about 2 axis at the same time. in three it cant

[Royston]

Can a body (say a ball) spin about multiple axes at the same time ?

 

In reality, of course it can, but it entirely depends on inertia i.e how the mass is distributed, so you're constricted in some sense (i.e an ideal sphere will be limited), but spin a ball, then kick it, so it's spinning in one direction, and another. Due to inertia, the axes of rotation are different.

 

Mathematically, a sphere can have as many axes of rotation as you wish.

 

in four dimensions it can rotate about 2 axis at the same time. in three it cant

 

That makes absolutely no sense whatsoever, a sphere is two dimensional for a start.

Edited June 16, 2009 by Snail

[granpa]

I'm sorry, I guess I just assumed that people were smart enough to understand that I meant a hypersphere.

[Royston]

I'm sorry, I guess I just assumed that people were smart enough to understand that I meant a hypersphere.

 

Even if you were referring to an n-sphere, your answer still makes no sense.

[granpa]

the fact that you dont understand it does not mean that it makes no sense.

[Royston]

the fact that you dont understand it does not mean that it makes no sense.

 

Well perhaps you could run us through it then granpa, AFAICS the question has been answered, but for some strange (troll like reason) you beg to differ.

 

So, for the viewers at home, explain how a hypersphere rotates.

[granpa]

rotation involves pairs of axes. thas why its best described as a bivector. when a 3d object spins about an axis the other 2 axes are interchanging. in 4 dimensions 2 axis interchange and the other 2 are unchanged.

 

in 3 dimensions it can be described by a psuedovector.

http://en.wikipedia.org/wiki/Pseudovector

[Royston]

rotation involves pairs of axes. thas why its best described as a bivector. when a 3d object spins about an axis the other 2 axes are interchanging. in 4 dimensions 2 axis interchange and the other 2 are unchanged.

 

in 3 dimensions it can be described by a psuedovector.

http://en.wikipedia.org/wiki/Pseudovector

 

Not very convincing grandpa, you've regurgitated something you've read, without understanding it.

 

Classically, rotation can be described with polar co-ordinates (personal preference), so your first statement is flawed i.e it doesn't require 'pairs of axes'. No mention of quaternions (at this level), how come ?

 

You can 'describe' rotation in any way you choose, be it a differential equation, a matrix, whatever...I'm getting the strong feeling you're just arguing for the sake of it.

[D H]

Granpa is right in this case. Rotation in 4-space is a different beast than rotation in 3-space. That should not be too surprising; rotation in 3-space is a very different beast than rotation in 2-space. It only takes one parameter to describe rotation in 2-space. It takes three parameters to describe rotation in 3-space, and six to describe rotation in 4-space.

 

A 2D square can rotate in 3-space like this animated GIF from http://eusebeia.dyndns.org/4d (All animated GIFs used in this post come from that site):

 

 

Nothing special here. That's just a rotation in 2-space viewed from above. However, a 2D square can also rotate in 3-space like this:

 

 

Another way of looking at rotation in 3-space is as a sequence of 2D rotations embedded in 3-space about a set of three pre-selected axes. The standard Euler sequence is a rotation about the z axis followed by a rotation about the x axis, followed by a final rotation about the z axis. Aerospace engineers tend to prefer a sequence of rotations about some permutation of the x, y, and z axes.

 

This can be extended to higher dimensions. Just as a 2D rotation embedded in 3-space is a rotation about a line, a 2D rotation embedded in 4-space is a rotation about a plane. There are six pairs of planes in 4-space, generating six separate primitive 2D rotations in 4-space. Here is a 2D portrayal (sorry, my screen is not four dimensional) of a 3D cube rotating about a plane in 4-space:

 

 

Denoting the axes of our 4D space as x, y, z, and w, this cube has zero w component and is rotating about the xw plane. (An alternate view: This cube is rotating in the yz plane.) Even simple 4D rotation adds something new. Here is a rotation about the yz plane (alternatively, a rotation in the xw plane):

 

 

Another consequence of the Euler rotation theorem in 3-space is that any rotation in 3-space can be viewed as a 2D rotation about some line in 3-space. This result does not extend to 4-space. Some rotations in 4-space can be viewed as a 2D rotation about some plane, but not all. Simultaneously rotating about a pair of planes in 4-space that have a common axis is equivalent to a single rotation about some other plane. Four dimensions adds a new twist: simultaneously rotating about a pair of planes that do not have a common axis. For example,

 

 

Here the cube is simultaneous rotating about the zw and xy planes (simultaneously rotating in the xy and zw planes). This is a Clifford rotation, or a double rotation. Double rotations are one of the weird things that can happen when objects in 4 space rotate.

[Royston]

Thanks DH, and sorry grandpa....I literally just walked out of a 3 hour physics exam, so being quite brash. I've only just got my head round solving partial differential equations, so sorry about that.

[proton]

I'm sorry, I guess I just assumed that people were smart enough to understand that I meant a hypersphere.

If someone didn't assume you were talking about a hypersphere it in no way means that they are not smart. For example; I myself initially thought you were referring to a three-dimensional object moving in a 4-dimensional space. How does that make me a not smart enough (for whatever we're supposed to me smart enough for)?

[defianceuvscien]

3 single times in the 3 single directions, is what this sounds like to me!:doh: