Block slides down hemisphere; time to leave surface?

[Dan6500]

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?

 

[Bystander]

You've obviously got an expression for angular velocity if you've determined "the where." Integrate it.

 

[Dan6500]

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]

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.