Before you are two balls, one solid and one hollow. They are to all appearences completely identical: same size, same weight, same outer material (though one might assume, correctly, that the hollow ball would need a higher-density material on the inside to make it the same weight).

Without breaking either of the balls, how can you easily determine which is hollow?

Assume that the material is solid enough that a hitting the side of the hollow ball will not result in any noticeable echo or vibrations.

Roll the balls down an incline. Make the balls don't spin while rolling down and rolls down "1-D" by making the paths narrow. The ball that rolls down FIRST will be the SOLID ball.

Let's look at some equations to see why this is true.

m=mass of the ball
r=radius of the ball
h=vertical height at the top of the incline
v=velocity at the bottom of the incline
w=rotational velocity at the bottom of the incline
I=moment of inertia

gravitational energy=mgh
translational kinetic energy=0.5*m*v^2
rotational kinetic energy=0.5*I*w^2
Q=any energy losses

w=v/r by since the ball is rolling down "1-D" and there is no spinning.

I=int(r^2*dm), dm is differential of mass
So basically I measures how far the mass is away from the center. So I is greater for the hollow sphere.

For spherical shapes, I=K*m*r^2, where K is a proportionality constant. It can be shown that K=2/5 for a solid sphere and K=2/3 for a spherical shell. In the given case of the hollow ball, the proportionality constant will be different since there are 2 different materials, but it will still be larger than that of the solid ball.

The energy loss for both balls is more or less the same since the balls are identical in almost all aspects.

 

Now writing a conservation of energy:

mgh=0.5*m*v^2+0.5*I*w^2+Q
mgh=0.5*m*v^2+0.5*K*m*r^2*(v/r)^2+Q
mgh=0.5*m*v^2+0.5*K*m*v^2+Q
mgh-Q=0.5*m*(1+K)*v^2
v=sqrt[(mgh-Q)/(0.5*m*(1+K))]

The velocity for the ball at the bottom of the ramp is greater for the solid sphere because it has a lower K. In fact, this equation shows that at any height, the solid sphere will travel at a faster velocity. Hence it will reach the bottom first.

Posted by np_rt on 2005-03-25 15:57:13