Opinions on rotating frames of reference

[Johnny5]

If you were going to teach someone about physics involving rotating frames of reference, where would you begin?

 

In other words where is the best place to begin?

 

I was thinking about starting off with the definition of angular momentum, and going from there.

 

L=R X P

 

From there you could go on to cover torque, and moments of inertia. The problem is I'm not sure the best place to start. The moment of inertia tensor is not for a beginner, so I need someplace else.

 

Any thoughts?

 

PS: My goal is to understand gyroscopes in particular, with a view towards quaternion algebra, and how it is used to avoid gimbal lock. Also, I want to tie the work here, to SO(3), which I am discussing with Tom Mattson in another thread.

 

There is just so much about rotating frames of reference out there, it seems difficult to compile it all, and make sense out of it, but that's what I am trying to do. So any suggestions at all would be most welcome.

 

Thanks

[Tom Mattson]

I would not start with any of the things you mentioned. I would start from simple kinematics. It would begin with the general expression for the derivative of a vector. If A is some (time-dependent) vector in some reference frame that has angular velocity w then dA/dt is given by:

 

dA/dt=(dA/dt)_rot+wxA.

 

Where:

 

(dA/dt)_rot is the derivative of the vector as observed in the rotating frame.

[Johnny5]

I would not start with any of the things you mentioned. I would start from simple kinematics. It would begin with the general expression for the derivative of a vector. If A is some (time-dependent) vector in some reference frame that has angular velocity w then dA/dt is given by:

 

dA/dt=(dA/dt)_rot+wxA.

 

Where:

 

(dA/dt)_rot is the derivative of the vector as observed in the rotating frame.

 

How do you come up with that formula? I believe I saw something resembling it at a NASA site' date=' I'm gonna check to see if i can find it again. The article was on the coriolis force.

 

I actually found it, its for centrifugal force, not coriolis force, here it is

 

NASA: Derivation of centrifugal force

 

Look down and you will see this:

 

(d /dt)_i = (d /dt)_r+ w x

 

That is Tom's expression practically.

 

How in the world do you come up with it is my question?

 

Let both operators operate on vector A, and you get:

 

dA /dt = (dA /dt)_rot+ w x A

 

So the LHS is the time derivative of vector A, in as seen from some external inertial frame of reference, and there is an equivalence.

 

So I guess what I am asking for, is to see a proof of the equivalence.

 

I read the article at NASA before, and didn't know where they came up with the formula above. I'm going to read it again.

[mezarashi]

Hmmm, I think the coordinate transformation here is simply the intuitive definition. I'll be drawing something on paint now to illustrate this. Though I hope I'm right. Rotational dynamics has never failed to stump me time and again

 

 

It would be more easy to comprehend if we refer to the change in vector A now as some velocity vector.

 

The absolute velocity of A with reference to some inertial frame is the velocity of A with respect to the body f in rotation, and the absolute speed of the body at which point A is at (w x r).