Challenge Physics Question: Sliding off a Hemisphere

Here’s a challenge physics question, which you can solve with a working knowledge of first-year college physics (even of the non-Calculus variety), but which you might find a bit tricky. I will post the answer in a week or two.

slideoffsphere

A small object slides frictionlessly from the top of a hemisphere of radius R. (The size of the object is much less than R.) It will slide along the hemisphere for a while, but at some point it will fly off of the hemisphere.

First question: at what position does the small object leave the sphere? You can give this answer by giving the angle θ shown on the picture above where the small object leaves the sphere.

Second question: what distance d does the object land away from the edge of the sphere?

Answers should be in terms of R, the radius of the sphere, and g, the gravity of earth.

If you want to visualize what’s happening, here’s a quick simulation I rendered using Blender’s physics engine.