Having a Ball

As a graduation present, Amy Bug (one of my Swarthmore physics professors) gave me an interesting little desk toy:

It's a rubber ball filled with a thick liquid, some glitter, and a smaller ball containing some LEDs and an accelerometer.  When the accelerometer detects that the ball has bounced, it starts blinking the LEDs for a few seconds.  I've found that a quick wrist-flick can also trigger them.  What I'd like to talk about here though is what happens when the ball spins:

I would have expected the glitter to tend toward the widest part of the ball, the center, but instead it fills the whole thing.

We'll assume that the thick liquid makes the force of gravity negligible, much like in the Neutral Buoyancy Laboratory used by NASA.  That just leaves the normal force that the sides of the ball apply to the glitter.  The radial component of this force will be related to the rotation of the ball by

where m is the mass of the glitter, r is the radius of the ball at a particular height, ω is the angular velocity of the ball's rotation, and N is the normal force.  Since the ball is spherical, we can express r in terms of the height, z:

where R is the radius of the ball at its center.  Using this, we can express the slope of the wall in terms of z as well:

This gives the direction of the normal force as

so putting this into our original equation and solving for N gives

Now that we have a general equation for N, we can find its z component:

Since we're assuming that the normal force is the only force on the glitter, we can write

Solving this differential equation with the initial condition z(0) = z'(0) = 0 gives

meaning a bit of glitter that starts at the bottom will oscillate from bottom to top with a frequency determined by the speed of the spinning.

There will be some amount of damping from the thick liquid, so if the ball kept spinning for long enough, the glitter would end up concentrated around the equator, but it appears that the damping is small enough that the ball stops spinning long before that happens.

Thanks, Amy!  It's a great thing to have for a fidgeter like me...