Dimensions of freedom in mass?

[arkain101]

I have a question. I would like others to add there input.

 

The question is, how many dimensions of freedom can an object contain momentum/kinetic energy.

 

Here is an example.

 

In a hypothetical universe only one object exists. This object is the shape of a ring.

 

Now, firstly it is impossible to say if the ring is moving in any direction. It also has no logical position. These are due to the lack of other objects existing to make those possibilities exist.

 

Next, imagine the ring is spinning like a wheel spins, such that, it looks as though it is at rest. This is a dimension of motion where the mass is moving in one dimension.

 

Lets say the ring is spinning at 100 revolutions per sec

 

Next, imagine the ring begin to start rotating in the perpendicular direction such that, it looks like a coin would look that has be "flipped" in the air. In this dimension / direction, the ring is rotating at 100 revolutions per second aswell. (in a vacum).

 

I believe a specific point on the ring covers a path that when seperated from the rest of the material looks like 3 dimensional figure 8 shape that depending on the velocity of perpendicular rotations, can morph from one perpendicular ring to another, by means of figure 8 shaped pathways.

 

This is another dimension of motion. However, does this secondary dimension of motion add to the total kinetic energy of the ring? Or does it compromise the velocities of the first type of spin, and have no additional effect?

 

So you can see what I am getting at. How many dimensions can a 3 dimensional object rotate to increase its rotational Ke? 1, 2, 3, or even 4?

 

Just by thinking about it, my intuition tells me the limit is 2 dimensions of freedom. But at the same time, any change in one dimension, affects the direction of motion in the other dimension, and should nulify the effect of increasing any energies.

 

 

 

So The correct answer should be 1 dimension.

 

What is the correct answer?

[arkain101]

To ask in a much more compact form, can you increase the angular momentum of a spinning object by giving it a wobble.

 

If you wobble it too much, will the object just begin rotating in the happy medium between the two tendancies?

 

The two tendancies being, perpendicular angles of rotation.

[CraigD]

The number of degrees of freedom of a n-dimensional rigid body subject to any possible acceleration is n(n+1)/2: n variables are needed to describe its spatial (translational) position, n(n-1)/2 its attitude (rotational position).

 

A body’s motion has the same number of degrees of freedom as its position and orientation (displacements). One degree of freedom describes its mass, and n(n-1)/2 its moments of inertia, so all of its momenta have [math]n^2+1[/math] degrees of freedom.

 

If the body has various symmetries, its degrees of freedom are reduced. A n-dimensional sphere, for example, has 0 degrees of freedom in its orientation, so its displacements or momenta only n degrees of freedom.

 

If the body is not rigid, its degrees of freedom, and thus the number of variables needed to describe its motion, are increased by up to a multiple of the number of independently moving rigid bodies constituting the nonrigid body. Non-rigid bodies are usually called systems.

 

The number of degrees of freedom of a n-dimensional rigid body not subject to any nonuniform acceleration is 2n: n variables are needed for its linear (translational) velocity, n-1 for the direction of its axis of rotation, and 1 for its rotational, or angular, speed. The reason that only n-1 variables are needed for the axis of rotation, a n-dimensional vector, is that it is a magnitude-less (unit) vector. By using azimuths, unit vectors can always be represented with 1 fewer variables than the number of spatial dimensions. This is a direct consequence of the mechanical concept of degrees of freedom.

 

If you’re only concerned with its angular speed, and axis of rotation, only n variable are needed. If you’re concerned only with its angular speed, only 1 variable is needed.

 

For a n-dimensional rigid body not subject to any uneven force, you can rotate your coordinate basis and ignore its translational velocity so that only 2 coordinate variable change for any point of the body.

 

If you are concerned with linear and angular momentum, a variable for mass and one for its moment of inertia are needed, so you can fully describe its momentum with 2n+2 variables.

To ask in a much more compact form, can you increase the angular momentum of a spinning object by giving it a wobble.

Giving wobble to a rigid body doesn’t increase its numbers of degrees of freedom, or necessarily change the magnitude of its angular momentum. It changes its axis of rotation and/or moment of inerta and/or angular momentum. In other words, your can always rotate your coordinates so that the body has no wobble.

In a hypothetical universe only one object exists. This object is the shape of a ring.

As arkain notes, being the only object in its hypothetical universe, this object has no meaningful translational position. Likewise, its axis of rotation is meaningless. It does have angular speed, and moment of inertia, so has 2 degrees of freedom.

 

Being the only body in its hypothetical universe, it can interact, so can’t accelerate, so as described above, you can chose a coordinate basis so that the body has no wobble.

 

If you assume that you can create ad-hoc angular accelerations, the body still has only 2 degrees of freedom. An acceleration may change only its angular speed, only its moment of inertia (by changing its axis of rotation relative to the body), both, or neither (by changing its axis or rotation while identically rotating the body).

 

The above is true regardless of the shape of the body. This make sense if you consider that the number of degrees of freedom can be interpreted as the number of numeric choices that can be made to define a system – in this case, a hypothetical, single body universe. No matter what shape the body is, it’s moment of inertia is given by a single number that depends on the body’s shape and its rotational axis, neither of which we can independently chose.