The other week an interesting question was posted on the OCC. A small ball is placed on top of a big one and rolls down. At some point the small ball will leave the surface of the large one, at what time after starting to move will this happen? The problem is that the motion is not uniform so the simple suvat equations don't apply however if we consider very small steps of motion then we can assume that acceleration is constant so the equations do apply. This would be rather difficult to do by hand but using a spreadsheet makes it do-able.

First we need to know at what angle the ball leaves the surface, this can be found by looking at the forces experienced by the ball.

The centripetal force is provided by he resultant of component of mg and N

mgcosθ -N = mv²/r

The ball will leave the surface when N = 0

So cosθ = v²/gr

When this takes place the ball will have fallen a distance h so applying the law of conservation of energy mgh = 1/2mv² but h=r- rcosθ. From this you can get that the ball will leave when θ = arcos(2/3)

To find the time taken to reach this angle is rather more complicated, we will consider the motion in steps of 0.1s.

• For the first step in the motion we can assume that the velocity is some initial value (0.01ms^-1), using this value we can calculate the displacement at constant velocity (acceleration =0 at the top).
• From this displacement we can find the angular displacement.
• Given θ we can find the tangential acceleration gsinθ
• Use the acceleration to find the displacement in the second step using the equation s = ut + 1/2 at² .
• Find the angular displacement.
• Add this angular displacement to the original value and repeat.
• Find the time taken for θ = arcos(2/3)

The answer for a sphere of radius 5m is about 5.1 s.

An alternative approach is to model the motion in Algodoo. You can determine the point at which the small ball loses contact by plotting the force vs time.

By the way Little and Large were a comedy duo from the 80's, not very funny but useful for trivia quizzes.

Here's the way I did it. Excel spreadsheet

The last columns are to eliminate values after the point where it leaves the sphere so the path could be plotted realistically.