Can someone help me with this physics question?

Question

I skier starts on top of a giant snowball. As the skier goes down, at what angle from vertical does the skier lose contact with the surface of the snowball?

Answer 1

The skier presumably maintains contact as long as there is some force of gravity pulling that skier toward the center of the snowball and that force overcomes the centrifugal force generated as the skier changes tangential velocity over the surface of the snowball.

Thus, the skier loses contact when C > W; where C = 1/2 mv^2/R = the centrifugal force and W = mg cos(theta) = the weight of the skier aimed to the center of the snowball and theta is the angle where theta = 0 puts the skier on the top of the snowball and theta = 90 puts her at the 9 o'clock position headed down.

The m's cancel out to give us 1/2 v^2/R > g cos(theta); thus 1/2 v^2/Rg > cos(theta) would give you the critical angle theta where centrifugal force begins to overcome gravity into the snowball. v = tangential velocity along the snowball's surface, R = the radius of the ball, and g = the acceleration towards Earth's center due to gravity.

If you assume the tangential velocity is due solely to the pull of gravity, then the tangential accleration over the surface of the snowball will vary as a = g sin(theta). Remember theta = 0 is on top the snowball; so there is no tangential acceleration on top. (We assume just a bit of a nudge here.)

Then, presuming the starting velocity is close to u = 0, v^2 = 2aS = 2 g sin(theta) S; where S = theta R, and theta is in radians so that 360 deg ~ 2pi radians. S is the distance along the arc of the snowball's surface.

So we have 1/2 v^2/Rg = 1/2 2aS/Rg = 1/2 2 g sin(theta)theta R/Rg > cos(theta) and theta > cot(theta) so your skier loses contact when theta > cot(theta)

Lesson learned: As so often happens when dealing with opposing forces, most of the factors cancel out.

Answer 2

From the vertical it would be 90, from the horizontal it would be 0.