A roll of supple paper of thickness x, wound tightly and initially of radius R>>x, has its free end attached so that when released it unwinds downward by gravity. Derive the equation that describes the dynamics of the unfurling roll, and determine its point of maximum velocity.

Just read your question and I assume it's in a vacuum. Also there is no tension forces in the roll.

It's maximum velocity should be at the end of however long a roll unfolds downward unobstructed. It should follow the law of gravity at whatever acceleration that gravity dictates. That's off the top of my head and I would enjoy any correction.

Just aman

aman, I think you have the general idea.

 

Can any of you now do the physics to "derive the equation that describes the dynamics of the unfurling roll?"

O.K. I'm having a little trouble so far cus I'm trying to put an equation together that incorporates the coreolis effect since the center of gravity is offset and I can't figure yet any centrifical acceleration with a decreasing radious with respect to time. It's gonna take some more work but it's a great question.

Just aman

You're kidding about the coriolis effect, aren't you? This isn't a sphere! Keep it simple, smartie.

The Coreolis effect on a spinning plate with a rolling ball in the center curves outward like an unwrapping roll of paper so it's not neccessarily a spherical phenomena and I was using it to help visualize the forces, but like you said, don't overengineer.

I found a good site with probably everything I need but it's been 17 years since I opened a physics book so it's going to take a little longer than I thought unless someone else jumps in.

reference

Just aman