# Question on Rotational Dynamics (Just understanding something, have solutions)

1. Jan 10, 2009

### Quantumpencil

1. The problem statement, all variables and given/known data

You have a rectangular frame with side lengths l and l', and a rigid rod attached to this frame at two points, r1 and -r1. The rod passes through the center of mass, and makes an angle theta with the horizontal. Then you begin to rotate the the frame around the rod with angular velocity $$\omega$$. What is the force exerted on the two points r1 and r2?

2. Relevant equations

$$\tau = r \times F$$
$$\ dL/dt = \omega \times L$$
$$\L = I\omega$$

Here omega denotes the angular velocity vector,

$$\omega = \omega (-cos \theta, sin \theta, 0)$$

I along the three principle axis = ml'^2, ml^2, ml'^2 + ml^2

3. The attempt at a solution

Using $$\L=I\omega$$ we obtain

$$\L=m\omega(l'^{2}cos \theta, l^{2} sin \theta, 0)$$

then, using

$$\ dL/dt = \omega \times L$$, we obtain

$$\tau = m \omega^{2} sin \theta cos \theta (l^{2}-l'^{2})$$

Which is equal to the torque. Torque is then equal to $$\tau = r \times F$$

the position of r1 as a function of theta is

$$\ r_{1} = l/2 (-1, tan \theta, 0)$$.

Now I have the solutions, but I don't really understand the rest of the solution; and I was hoping someone could help clarify.

According to the solution guide:

Since the rod is frictionless, we want a force perpendicular to the rod (why?) of the form $$\ F_{1} = f (sin \theta, cos \theta, 0)$$ (Why!?)

Which satisfies $$\2r_{1} \times F_{1} = (fl/cos \theta)z = \tau$$(WHY!?)

z denotes the unit vector.

Then we just set it equal to the expression I understand for torque, solve for f (the magnitude of the force), and we have the solution.

The steps with a "Why" are not very clear to me, and I'd really love to change that.

2. Jan 11, 2009

### tiny-tim

Hi Quantumpencil!

A frictionless body can only exert a force normal to its surface.

And cos + sin*tan = 1/cos

3. Jan 11, 2009

### Quantumpencil

Re: NEW Question on Rotational Dynamics (Would like some opinions on a problem)

Right, ok. I feel a little ridiculous for not thinking of that... It's just like a flat surface or whatever, the force being applied is a normal force just like in all the Mechanics problems from HS. Ugh -_-

So I'm trying to construct some good follow-up questions to the set up posted here.

I thought of one involving the motion of the object if the rod is quickly yanked when l=/=l' and l=l', but I'm afraid it's too elementary. So here's my next stab.

9 - 3: The diagram shows our favorite light rigid rectangular frame with masses at the corners (that we have been talking about in lecture). But instead of letting it float in space, we drill holes through the frame at r1 and -r1 and put a frictionless rod through the holes as shown. Now we set the frame in rotation about the fixed rod with angular velocity ω with its center of mass fixed and the masses above the rod (in the diagram) going in the negative z direction, into the plane of the paper.

Follow up - Assume that immediately after the impulse, the particle at r1 making up the rod magically passes through the frame, and the frame is attached only at - r1. Afterwards, the frame:

1: will continue to rotate with angle velocity ω, because the body is rigid

2: will precess wildly around the axis of the rod, because there is now net torque on r1.

3: will precess until settles at a new angular velocity, ω'.