Confused on a mechanics problem

[blendecho]

Hello,

 

I'm having a bit of difficulty with this one mechanics problem, and I'd appreciate any pointers. So without further ado...

 

"A thin semicircular hoop of radius R and mass m rocks back and forth without slipping on a rough surface. Find the period of oscillation for small amplitude oscillations, reducing to lowest order in oscillation angle theta. Then find the period if the surface is frictionless."

 

Oh, and the center of mass is 2R/pi below the center of curvature.

 

Thanks in advance.

[swansont]

Step 1 is draw a diagram. There must be a restoring force when it's been perturbed, and you need to find this. The "small amplitude oscillations" is usually code for "you can use ^{[math]sin\theta \approx \theta [/math]} " (must use radians for that)

[blendecho]

Yeah, that much is pretty clear. I'm mostly stuck on, in the second part, getting the lever arm to the center of mass, which is the only torque about the contact point. I guess that's a bit of a geometry problem. In the first part I know what conditions hold but I'm not quite sure how to use them.

[swansont]

If the contact point makes the angle theta with the center of the circle, and the COM makes a different angle (phi) connecting the contact point to the COM, then you can use some geometry (alternate interior angles are equal) and the law of sines to relate the two. If the resulting moment arm is r,

 

[math]sin\phi /(2R/\pi) = sin\theta/r[/math]

[blendecho]

I see that, but I still don't see where that gets me as I don't know phi either.

 

What I'm trying to do is write down torque as a function of sin(theta), so really all I want is to find the lever arm since the sin(theta) kind of takes care of itself for small theta.

[swansont]

You get the torque in terms of phi, and then use the identity to put it in terms of theta. Then you have an equation solely in terms of theta that you can solve.

[blendecho]

It ended up working pretty well to just consider torques about the center of mass for the rolling case, but to consider torques around the center of curvature for the slipping case. Nasty geometry can really complicate things though.

 

As a side note the ratio of the two periods is rt(pi/2-1) which is kind of cool.