Particle rolling around inside of hemispherical bowl

[JohnDough]

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 v_o. It travels halfway around the sphere and reaches point B, which is a vertical distance h below A, with a velocity v_f. Point A is a radial distance of r_o 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 v_f.

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

2. Homework Equations

Conservation of Momentum, Energy
r² + h² = r₀²

3. The Attempt at a Solution

L_o=L_f
mr_ov_o=mrv_fcosθ
θ=arccos((mr_ov_o)/(mrv_f))=arccos((r_ov_o)/(rv_f))
KE_o+PE_o=KE_f
1/2 mv_o²+mgh=1/2 mv_f²
v_o²+2gh=v_f²
√(v_o²+2gh)=v_f
θ=arccos((mr_ov_o)/(mrv_f))=arccos((mr_ov_o)/(mr√(v_o²+2gh)))

 

[haruspex]

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 v_f." 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?