Question

A solid uniform sphere and a thin-walled, hollow sphere have the same mass M and radius R. If they roll without slipping up a ramp that is inclined at an angle β above the horizontal and if both have the same vcm before they start up the incline, which object reaches the greater height, or do both of them reach the same height?

Answer 1

Answer:Hollow sphere

Explanation:

Given

same mass for solid and hollow sphere

same $v_{cm}$ before they start up incline

Moment of inertia of solid Sphere

$I_1=\frac{2}{5}Mr^2$

Moment of inertia of hollow sphere

$I_2=\frac{2}{3}Mr^2$

Conserving Energy at bottom and top point for solid sphere

kinetic energy +Rotational Energy=Potential energy

$\frac{1}{2}Mv_{cm}^2+\frac{1}{2}I\omega ^2=mgh_1$

for pure rolling $v_{cm}=\omega r$

$\frac{1}{2}Mv_{cm}^2+\frac{1}{2}\times \frac{2}{5}Mr^2=Mgh_1$

$\frac{7}{10}Mv_{cm}^2=Mgh_1$

$h_1=\frac{7v_{cm}^2}{10g}$

For hollow sphere

$\frac{1}{2}Mv_{cm}^2+\frac{1}{2}\times \frac{2}{3}Mr^2=Mgh_2$

$h_2=\frac{5v_{cm}^2}{6g}$

therefore height gained by hollow sphere is more