A solid sphere ( ) and a hollow thin-walled cylinder ( ) have identical masses and radii

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems

Problem

A solid sphere (

k = \tfrac{2}{5}

) and a hollow thin-walled cylinder (

k = 1

) have identical masses

M

and radii

R

. Both are released simultaneously from rest at the same height on an incline of length

10\,\text{m}

that makes an angle of

30^{\circ}

with the horizontal. Each object rolls without slipping, and resistive forces can be neglected. (a) Express the time

t

required for an object with general moment-of-inertia constant

k

to reach the bottom of the incline in terms of

g

k

, and the geometric parameters of the incline. (b) Using your expression from part (a), determine the ratio

t_{\text{hollow}}/t_{\text{sphere}}

for the two objects described. (c) Provide a qualitative explanation for why the ratio you obtained differs from unity, making explicit reference to the energy distribution during the motion.

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