# Block slides down hemisphere; time to leave surface?

## Main Question or Discussion Point

A block is at rest at the top of a frictionless hemisphere of radius r. It is slightly disturbed at starts sliding down. I already know where it will leave the surface (height = 2r/3). My question is, WHEN will it leave the surface?

## Answers and Replies

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Bystander
Science Advisor
Homework Helper
Gold Member
You've obviously got an expression for angular velocity if you've determined "the where." Integrate it.

I've got v as a function of theta: v = (2gr(1-cos(theta))^0.5. So now I integrate with respect to TIME? How?

Philip Wood
Gold Member
Expanding what Bystander said (in case you're not familiar with angular velocity)
$$v=\frac{ds}{dt}=r\frac{d\theta}{dt}$$
in which ds is an increment of arc length. r is, of course, a constant.
You can substitute this value for v into your equation, then separate variables.
If I'm not interested in the challenge of the integration I sometimes use the Wolfram on-line integrator.