- #1

Cepterus

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## Homework Statement

A mass point is sliding down from the top of a hemisphere, with an initial speed of zero.

I want to determine the angle ##\theta## at which the mass point will lift off the surface of the hemisphere.

## Homework Equations

## The Attempt at a Solution

We can determine the speed ##v## at any given angle ##\theta## by using the conservation of energy:

$$\begin{align*}E_{\rm{kin}}+E_{\rm h}&=E_{\text{h,0}}\\

\frac12mv^2+mgr\cos\vartheta&=mgr\\

\Rightarrow v&=\sqrt{2gr(1-\cos\vartheta)}\end{align*}$$ and geometrical considerations lead to horizontal and vertical components of ##v_x = v\cos\vartheta## and ##v_y = v\sin\vartheta##.

I need some characterization of the spot on the surface at which the lift-off is going to occur, but I don't have any good ideas.