Two rolling balls

This problem was suggested by Toby Crisford.



Two balls of equal radius and mass, free to roll on a horizontal plane, are separated by a distance L large compared to their radius. One ball is solid, the other hollow, and they are attracted by a mysterious force. How far will the solid ball roll before it collides with the hollow ball?

Solution by Michael A. Gottlieb

The acceleration of a rolling object with mass M, radius R, and moment of inertia kMR^2 (0<k<1) under force F (parallel to the rolling plane) is F/(k+1)M.  For the solid sphere k=2/5 and for the hollow one k=2/3. F and M  are the same for both balls, so the ratio of their accelerations must be constant and equal to (1+2/5)/(1+2/3) = 21/25. This must also be the ratio of the distances the balls traverse in any given period. (Note: this is independent of the specific properties of the force.) So, the solid ball goes (25/46)L and the hollow ones goes (21/46)L.


It is interesting to note that you can guess the answer to this problem using the following line of reasoning:  Call the distances traveled d1 and d2 (with d1+d2 = L). The initial conditions and forces are symmetrical on the two balls, which are identical, except for the dimensionless factors 'k' in their moments of inertia, kMR^2. So, the only possible numbers that can appear in the answer are k1, k2, and L (with, let us say, k1 for the solid ball and k2 for the hollow one). Obviously then, the answer has to be of the form d1= f(k1,k2)*L, and by symmetry we expect d2 = f(k2,k1)*L, such that  f(k1,k2)+f(k2,k1)=1. We know that when k1=k2, we must have d1=d2. We also know that the ball with the smaller moment of inertia will accelerate faster and thus go farther. So a first guess might be d1/d2 = k2/k1. However, this can be ruled out by considering what happens when, for example, k2 is fixed and k1 becomes arbitrarily small: d1 becomes arbitrarily large, which makes no sense, since it must be less than L. A reasonable second guess is d1/d2 = (k2+c)/(k1+c) for some constant c. Since 0 < k1,k2 < 1, c can not be much larger than 1, otherwise you would always have d1 ~= d2. In fact, if c>1, then you might say that the constant has more influence over the ratio d1/d2  than do k1 and k2, which doesn't seem right. So c = 1 is a reasonable guess, and this implies the correct solution:

   d1 = L/(1+(k1+1)/(k2+1)),
   d2 = L/(1+(k2+1)/(k1+1)).