Amazing Math Applications: Movement Inside a Sphere

Modeling how a particle moves on the inside of a sphere

I never fail to be amazed at how — in a few lines of formulae— a complex looking real life situation can be modelled with math. Consider the following problem:

A particle of mass m rests at the bottom inside a hollow smooth sphere with center O. It is then projected along the inside surface until it reaches a point P in the upper hemisphere where it leaves the surface. It then moves through the hollow space inside the sphere until it hits the inner surface again at a point C. Find the minimum angle that the point P must make with the vertical in order to be sure that the point C is in the lower hemisphere.

This sounds like it would be very hard to model with math, but in fact we can model it reasonably easily and the answer is really intuitive. Stick with me as I break it down.

Laying out the problem

Let’s start by drawing a picture of the situation. Here’s my modest attempt.

We see our point P in the upper hemisphere at which the particle leaves the surface. We have defined the angle that this makes with the vertical as θ. At this point we call the velocity with which the particle leaves the surface as v. We mark C as the…