No Friction!

Classical Mechanics Level pending

A particle of mass \(m\) is initially at rest on top of a solid hemisphere, also of mass \(m\). An infinitesimal push is given to the particle so that it begins to slide down the hemisphere. The ground, the hemisphere, and the particle are made of ice with negligible friction. The particle loses contact with the hemisphere at an angle \(\theta\) to the vertical. What is \(\cos \theta\)?

Imagine the hemisphere as a globe cut down the middle and then placed flat on the ground.

This problem is taken from Harvard's Problem of the Week.

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