Hi Don,
This is more of a physics question than a math question, but I hope the
following hints will help. I won't answer the question completely
here, but if you have trouble, please write back and I will try to
go through it in more detail.
One of the key ideas in this problem is the concept of moment of inertia,
usually expressed by the letter I (capital-I). First, the ping-pong ball
(with mass m) is lighter than the golf ball [which we will assume
spherical (i.e. ignore all the little dimples), with mass M]: m < M.
Let both balls have radius R. So far, we're just naming the quantities
that could be important.
The moment of inertia of the ping-pong ball is approximately m*R^2. This
is because all of the bits of celluloid that make up the ball are located
at a distance R from the center--the ball is completely hollow. Now, I
don't know exactly what is inside a golf ball, but from TV commercials, I'm
pretty sure that golf balls are not completely hollow. So the moment of
inertia of the golf ball is a*M*R^2, where a is some number such that
0 < a < 1.
You will have to use this information when you write down the dynamical
equations that govern how the balls roll down the incline. This will
involve the acceleration down the plane due to m*g*sin(theta), but don't
forget the frictional force at the contact point, which causes rotation
via a torque.
Give that a try, and see what you come up with. But let me also mention
another method (not a hint) that I sometimes like to use when I am not sure
exactly what will happen: "do the experiment." Maybe you can't find
two balls that exactly satisfy what is stated in your problem, but you
might be able to do a closely-related experiment, just to get an idea of
what might happen. Take two cans of soda (or beer, or whatever). Drink
the contents of one. Now these two objects have the same radius, but
different masses and certainly the mass is distributed differently--they
have different moments of inertia (which can is more like the ping-pong
ball and which can is more like the golf ball?). Now what happens if you
roll these two cans down a slope? There are a couple important differences
between the can experiment and the balls (they have different formulas
for the moment of inertia) and the full soda can has some extra internal
degrees of freedom (because the liquid can slosh around), but the
experiment is fun to do, just the same, and it might tell you what will
happen in the case with the balls.
Good luck, and please write back if you need any additional help.
- Doctor Douglas, The Math Forum