FYKOS Year 39 Series 3 Problem 1 — (un)stable toilet paper · FYKOS 39 #39.3.1
Physics · mechanics · thermodynamics
Problem
Danka is interested in how the stability of a toilet-paper roll changes
as part of the paper is used. She therefore took a new roll containing
N=120 sheets and measured its parameters. She found that the roll
has height h=9.5cm, diameter d1=12.5cm, and mass $m_1 =
139\,\mathrm{g}.Thecardboardtubeonwhichthepaperiswoundhasadiameterd_0 = 3.5\,\mathrm{cm}andmassm_0 = 4\,\mathrm{g}$. By what factor does the
potential energy required to tip the roll decrease when only one-third
of the paper remains compared to the initial state? Assume the layers of
paper are wound uniformly and tightly, without volumetric deformation.