This week I changed the toilet paper in my bathroom, and it got me thinking about the rotational properties of a roll – When you try to spin something, you have to fight against rotational inertia, which depends both on mass, and how far from the center it's distributed. That's why figure skaters speed up their spin when they pull their arms in:
What made me curious is that as the roll turns, the radius decreases, which lowers both the inertia and the torque that spins it. Do these balance out? I figured I'd try modelling what happens if you drop a roll while holding one end.
Since we're only interested in the roll rotating in the usual way, we can consider things in 2D:
We can find the rotational inertia by thinking of the roll as a disk with radius r(t), minus a smaller disk of radius r_i:
where ρ is the mass-density of the roll. Now we can set up a bunch of other equations that we need. By holding on to the end, we're applying a torque:
where α is the angular acceleration of the roll. If we suppose each sheet has thickness T, we can relate the radius to the angle the roll has turned through:
Finally, we can integrate to connect the angle to the distance the roll has dropped:
Combining the torque and radius equations gives a differential equation:
This is fairly simple to solve by integration:
but I'm having trouble getting a plot. I guess this is another week with complex equations leading to unsatisfying results. Thanks for sticking through :)
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