Answer
The total kinetic energy of the hollow cylinder is $Mv^2$
The total kinetic energy of the solid sphere is $\frac{7}{10}Mv^2$
The total kinetic energy of the solid cylinder is $\frac{3}{4}Mv^2$
We can rank the objects by their kinetic energies, from smallest to largest:
$solid~sphere \lt solid ~cylinder \lt hollow~cylinder$
Work Step by Step
We can find the total kinetic energy of the hollow cylinder:
$KE = KE_{tr}+KE_{rot}$
$KE = \frac{1}{2}Mv^2+\frac{1}{2}I~\omega^2$
$KE = \frac{1}{2}Mv^2+\frac{1}{2}(MR^2)~(\frac{v}{R})^2$
$KE = \frac{1}{2}Mv^2+\frac{1}{2}Mv^2$
$KE = Mv^2$
We can find the total kinetic energy of the solid sphere:
$KE = KE_{tr}+KE_{rot}$
$KE = \frac{1}{2}Mv^2+\frac{1}{2}I~\omega^2$
$KE = \frac{1}{2}Mv^2+\frac{1}{2}(\frac{2}{5}MR^2)~(\frac{v}{R})^2$
$KE = \frac{1}{2}Mv^2+\frac{1}{5}Mv^2$
$KE = \frac{7}{10}Mv^2$
We can find the total kinetic energy of the solid cylinder:
$KE = KE_{tr}+KE_{rot}$
$KE = \frac{1}{2}Mv^2+\frac{1}{2}I~\omega^2$
$KE = \frac{1}{2}Mv^2+\frac{1}{2}(\frac{1}{2}MR^2)~(\frac{v}{R})^2$
$KE = \frac{1}{2}Mv^2+\frac{1}{4}Mv^2$
$KE = \frac{3}{4}Mv^2$
We can rank the objects by their kinetic energies, from smallest to largest:
$solid~sphere \lt solid ~cylinder \lt hollow~cylinder$
