Why, on some toilet paper dispensers, can I get a long piece of toilet paper without tearing if the roll is fat, but when the roll has been nearly used up, the paper inevitably breaks too soon, giving only short pieces? Why is the opposite true for other dispensers?

I tried to think about rotational inertia and torque experienced by the toilet paper, where in the lever arm is the radius of the toilet paper, and the applied perpendicular force would be the force applied from the tearer. But if I stick with this, I end up having the fat roll tearing shorter pieces than the roll nearly used up. Am I missing something here?

Sounds like you spend too long on the toilet pondering the toilet paper dispenser. If the toilet paper dispenser was operated by some kind of electrical device, I could understand your question, but it isn't, it is completely manual and intermittent, spring loaded and completely beyond our control. It has a mind of its own and if we are not careful, toilet paper dispensers will take over the world!

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