We did this as a final problem in our study of rotational energy. After working through a series of long equations from energy and kinematics, we discovered the final answer is surprisingly elegant:

UPDATE 12/27/2010: Thanks to Dan Fullerton’s class for catching our error. They analyzed the problem using a torque approach and came up with:

I double-checked our energy approach and now get the same answer as Dan. We must have made an algebra mistake somewhere. This is why I love physics — there is more than one way to solve a problem!

H is the drop height of the free-falling roll, h is the drop height of the unrolling roll, r and R are the inner and outer radii of the unrolling roll.

Using large rolls of paper towels, we tested our prediction. Here’s the result, captured in slow motion:

(Despite our mistake, the demo still works. From the video, it seems the students did not release the rolls simultaneously. Perhaps this compensated for our algebra error.)

They were just a tad excited when it worked. And yes, that class is all boys, much to my dismay.

What’s your favorite activity or demo for rotational motion?