Question

A solid sphere, a holIow sphere and a disc of same mass and same radius are released from a rough inclined plane. All of them rolls down without slipping. On reaching the bottom of the plane, Match the two columns

Answer 1

a) $t=\sqrt{\frac{2S}{a}}$ and $a=\frac{g\sin \theta }{1+I/m{R}^2}$
$I$ of hollow sphere is maximum. So, a is minimum and therefore t is maximum
b) $K_{Total}=mgh$ for all
c) and d)$\frac{K_{rot}}{K_{tr}}=\frac{2}{5}$ for solid sphere
$\Rightarrow (\frac{K_{rot}}{K_{total}}{)}_{solid}=\frac{2}{2+5}=\frac{2}{7}$
$\frac{K_{rot}}{K_{tr}}=\frac{2}{3}$ for hollow sphere
$\Rightarrow (\frac{K_{rot}}{K_{total}}{)}_{hollow}=\frac{2}{2+3}=\frac{2}{5}$
$\frac{K_{rot}}{K_{tr}}=\frac{1}{2}$ for disc
$\Rightarrow (\frac{K_{rot}}{K_{total}}{)}_{disc}=\frac{1}{1+2}=\frac{1}{3}$
Hence,
$(\frac{K_{rot}}{K_{total}}{)}_{solid}<(\frac{K_{rot}}{K_{total}}{)}_{disc}<(\frac{K_{rot}}{K_{total}}{)}_{hollow}$
$\Rightarrow (\frac{K_{tr}}{K_{total}}{)}_{solid}>(\frac{K_{tr}}{K_{total}}{)}_{disc}>(\frac{K_{tr}}{K_{total}}{)}_{hollow}$