A hoop, a solid cylinder, a solid sphere, and a thin, spherical shell each have the same mass of $3\mathrm{.}87$ kg and the same radius of $0\mathrm{.}253$ m$.$

$($a$)$

What is the moment of inertia $($in kg $$ m$2)$ for each object as it rotates about its axis?

hoop kg $$ m$2$solid cylinder kg $$ m$2$solid sphere kg $$ m$2$thin, spherical shell kg $$ m$2$

$($b$)$

Each object is rolled down the same ramp, each starting from rest at the same initial height. Each rolls without slipping. Rank the translational speed at the bottom of the ramp for each object from highest to lowest.

solid cylinder $>$ thin spherical shell $>$ solid sphere $>$ hoopsolid sphere $>$ solid cylinder $>$ thin spherical shell $>$ hoop$$hoop $>$ solid cylinder $>$ solid sphere $>$ thin spherical shellthin spherical shell $>$ solid sphere $>$ solid cylinder $>$ hoop

$($c$)$

As in part $($b$),$ each object is rolled down the same ramp, each starting from rest at the same initial height. Each rolls without slipping. Rank the rotational kinetic energy at the bottom of the ramp for each object highest to lowest.

solid cylinder $>$ thin spherical shell $>$ solid sphere $>$ hoophoop $>$ thin spherical shell $>$ solid cylinder $>$ solid sphere$$hoop $>$ solid cylinder $>$ solid sphere $>$ thin spherical shellthin spherical shell $>$ solid sphere $>$ solid cylinder $>$ hoop