Conservation of energy related question

[endeavor]

Homework Statement

A skier of mass m starts from test at the top of a solid sphere of radius r and slides down its frictionless surface. (a) At what angle $$\theta$$ will the skier leave the sphere? (b) If friction were present, would the skier fly off at a greater or lesser angle?
http://img158.imageshack.us/img158/2091/chp8pro24pu5.png

Homework Equations

Conservation of energy (because this problem is from that chapter)

The Attempt at a Solution

The skier will leave the sphere if the normal force becomes zero. So radially:
$$mg \cos \theta - N = \frac{mv^2}{r}$$
$$mg \cos \theta = \frac{mv^2}{r}$$
$$v^2 = rg \cos \theta$$
Then, taking the potential energy reference to be the height when the skier leaves the sphere,
$$E_i = E_f$$
$$K_i + U_i = K_f + U_f$$
$$0 + mg(r - r \cos \theta) = 1/2 m (rg \cos \theta) + 0$$
But this doesn't work if I try to solve for theta!

 

[lippyka]

What am I missing?The solution is that we assume the skier does not slip radially, so v=\sqrt{rg \cos \theta}. Thenmg \cos \theta = \frac{m(rg \cos \theta)}{r}\cos \theta = 1\theta = 0.If friction were present, the skier would fly off at a lesser angle, since the normal force would be less to counteract the skier's centripetal acceleration.

 

[m2003]

I would like to point out that your attempt at a solution is on the right track, but there are a few errors in your equations. First, the normal force should be equal to the centripetal force, not the other way around. This is because the normal force is responsible for providing the necessary centripetal force for circular motion. So the correct equation should be:

mgcosθ = mv²/r + N

Also, the potential energy reference should be at the bottom of the sphere, not when the skier leaves the sphere. This is because the skier has potential energy at the top of the sphere due to its height, and as it slides down, this potential energy is converted into kinetic energy. So the correct equation should be:

mg(r-r cosθ) = 1/2mv² + 0

From here, you can solve for θ using the conservation of energy equation. As for the second part of the question, if friction were present, the skier would fly off at a lesser angle. This is because friction would act against the motion of the skier, causing it to slow down and have less kinetic energy. This would result in a smaller angle of projection off the sphere.