Problem — Ranking four rolling objects

Problem

Four objects of equal mass $m$ roll without slipping down an identical incline, starting from the same height: (A) solid sphere, $I = \tfrac{2}{5}mR^2$; (B) solid cylinder, $I = \tfrac{1}{2}mR^2$; (C) thin ring, $I = mR^2$; (D) solid sphere of a different radius, $I = \tfrac{2}{5}mR^2$. Rank the times taken to reach the bottom, from smallest to largest. (Hint: $a = g\sin\theta/(1 + I/(mR^2))$ and does not depend on $R$.)

Solution

The acceleration is $a = \dfrac{g\sin\theta}{1 + I/(mR^2)}$: the larger $I/(mR^2)$, the smaller $a$, the longer the descent time. The radius does not appear, so the size of the object is irrelevant.

Factors $I/(mR^2)$: solid spheres (A and D) $= \tfrac{2}{5}$ → maximum acceleration → shortest time; solid cylinder (B) $= \tfrac{1}{2}$ → intermediate; ring (C) $= 1$ → minimum acceleration → longest time. The two spheres have the same factor, hence the same time (the radius doesn’t matter). $\ev{\text{A} = \text{D} < \text{B} < \text{C}}$

Connections

Topics: Dinamica rotazionale Concepts: Rotolamento · Momento d’inerzia Skills: Ragionamento di ranking Objects: Piano inclinato