Particle rolling around inside of hemispherical bowl

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1. Homework Statement

I have a hemispherical bowl in which I roll a small particle around the edge, starting from the top at point A with a velocity vo. It travels halfway around the sphere and reaches point B, which is a vertical distance h below A, with a velocity vf. Point A is a radial distance of ro from the vertical centerline and point B is a radial distance of r from the vertical centerline. There is no friction. The goal is to solve for the angle, θ, between the horizontal and the velocity vf.

Diagram: http://i.imgur.com/57qgEHI.png

2. Homework Equations

Conservation of Momentum, Energy
r2 + h2 = r02

3. The Attempt at a Solution

Lo=Lf
mrovo=mrvfcosθ
θ=arccos((mrovo)/(mrvf))=arccos((rovo)/(rvf))
KEo+PEo=KEf
1/2 mvo2+mgh=1/2 mvf2
vo2+2gh=vf2
√(vo2+2gh)=vf
θ=arccos((mrovo)/(mrvf))=arccos((mrovo)/(mr√(vo2+2gh)))
 
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Answers and Replies

  • #2
haruspex
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You mean conservation of angular momentum, right? That is only going to be valid about the vertical centreline, as you appear to have appreciated. (You understand why, right?) But there may be some confusion over the angle theta. The OP says it's "between the horizontal and the velocity vf." To me, that is not the same as saying it's between the horizontal tangent to the sphere and the velocity vf, yet that's what your angular momentum equation implies to me. The diagram does not make it clear.
Have you been told your answer is wrong, or are you merely seeking corroboration before submitting it?
 

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