Solving Goldstein Problems: Point Mass vs. Hoop on Fixed Hemisphere

[robb_]

I just solved two of Goldstein problems. let me give you the gist.
1. A point mass is on a fixed hemisphere under the influence of a g field.
2. A hoop is on a fixed hemisphere under the influence of a g field.

I have found the equations of motion, etc... no probs there.
I found that the point mass will leave the sphere at a smaller angle than the hoop. Here the angle is measured w.r.t the vertical.
I would not have guesed this and am wondering if anyone has a good conceptual explanation for this. thanks

 

[Doc Al]

I presume the hoop is rolling without slipping down the hemisphere (there is friction present).

In any case, gravity provides the centripetal force. But at some point the object is going too fast for the radial component of gravity to provide enough force to keep it in contact with the sphere. But note that the hoop rolls as well as translates--thus it takes longer to build up enough speed. (It's change in gravitational PE must support rotation as well as translation.)

 

[nrqed]

I just solved two of Goldstein problems. let me give you the gist.
1. A point mass is on a fixed hemisphere under the influence of a g field.
2. A hoop is on a fixed hemisphere under the influence of a g field.

I have found the equations of motion, etc... no probs there.
I found that the point mass will leave the sphere at a smaller angle than the hoop. Here the angle is measured w.r.t the vertical.
I would not have guesed this and am wondering if anyone has a good conceptual explanation for this. thanks

I would say that this is reasonable given that at any given angle, the point mass is moving faster than the hoop. The reason is of course th emoment of inertia of the hoop, some energy goes into kinetic energy of rotation, leaving less for kinetic energy of translation. In the case of the point mass, all the energy goes into kinetic energy of translation.

My two cents.

Patrick

 

[robb_]

The hoop rolls without slipping as stated in the problem. (Funny though, it turns out that friction will not be large enough to keep it from slipping after a certain angle which can be less than the angle at which the hoop leaves the surface.)
So if I follow the reasoning above, I would guess that for a frictionless hemisphere the angles would be equal?

 

[Doc Al]

The hoop rolls without slipping as stated in the problem. (Funny though, it turns out that friction will not be large enough to keep it from slipping after a certain angle which can be less than the angle at which the hoop leaves the surface.)

Right--slipping complicates things, but doesn't change the fact that at any given angle the hoop will be moving slower than the point mass.

So if I follow the reasoning above, I would guess that for a frictionless hemisphere the angles would be equal?

That's what I'd say. Without friction the hoop will not rotate, so its speed will match that of the point mass.

 

[robb_]

Thank you greatly.