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This really bothers me as I think I should know the answer...

I pushed a toilet paper roll so it slowly rolled of a five centimeter edge. On the next surface it had gained a lot speed. But why?

It is simply a question of gravitational energy being converted from vertical to horizontal motion? If that is it, what is the principle that governs that? Had I dropped (a perfectly round) toilet paper roll on a (perfectly flat) surface perfectly vertical it had not started to move horizontally. Does it then follow that a falling object with horizontal motion is able to convert some of the energy from the impact into horizontal movement while the same object falling straight down cannot.

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  • $\begingroup$ Hint: You didn't say, but I can confidently guess you had it unrolling. $\endgroup$ Oct 11, 2019 at 19:01
  • $\begingroup$ What exactly do you mean by a "five centimeter edge"? $\endgroup$
    – Bob D
    Oct 11, 2019 at 20:43

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I tried this on my kitchen counter top. The angular velocity of the roll seems to increase as it comes off the edge of the counter top. (@dmckee it didn't unroll cause I taped it). Note: my counter top is beveled at the edge, not a sharp corner as shown in the figure below).

I could be wrong, but perhaps the explanation is as follows:

While rolling on the counter top the upward reaction force of the counter top equals the gravitational downward force on the roll. The reaction force acts through the center of gravity of the paper towel roll. There is no net torque about the CG. See the left figure below.

As the roll of paper towels start to fall off the edge, but is still in contact with the edge, the reaction force of the counter top no longer acts through the center of gravity of the roll of paper towels. See the figure to the right, below. There is now a net torque acting on the CG of the roll that causes an increase in the angular velocity of the roll. When it contacts another horizontal surface just below it will continue on with a greater angular velocity than the original.

Hope this helps.

enter image description here

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  • $\begingroup$ There would be another effect, as far as I can see it. The force f would not be expected to remain vertical, but would gain a component to the right. Part of f would make a couple with gravity to create torque, but part of it would accelerate the object to the right. If contact was entirely without friction, the object would still accelerate to the right. Depending on whether its rightward velocity was slow enough to keep in in contact with the table as it falls off the edge. And that would also affect the torque. $\endgroup$ Oct 11, 2019 at 22:37

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