Amazing Math Applications: Movement Inside 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 earliest point at which the particle can hit the surface for it to be in the bottom hemisphere. Finally, we assume our sphere has unit radius. We could call the radius r, but it will make our manipulations easier if we just give it a unit radius, and doing so won’t impact the answer.
Examining the forces at the point P
When the particle is in contact with the surface, a reaction force is exerted in the direction of the center of the sphere. marked as R in the diagram above. Also gravity acts downwards and is equal to mg. Therefore the component of the force of gravity that acts inwards to the center of the sphere can be resolved as mgcosθ. Using Newton’s second law, we can say that