A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed…
A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $v$. The sphere and the cylinder reaches upto maximum heights $h_1$ and $h_2$, respectively, above the initial level. The ratio $h_1: h_2$ is $\frac{n}{10}$. The value of $n$ is______.
Solution
Gain in P.E. = Loss in K.E.
$\mathrm{mgh}=\frac{1}{2} \mathrm{mv}^2\left(1+\frac{\mathrm{K}^2}{\mathrm{R}^2}\right)$
$\mathrm{h} \propto 1+\frac{\mathrm{K}^2}{\mathrm{R}^2}$
$\frac{\mathrm{h}_1}{\mathrm{~h}_2}=\frac{1+\frac{2}{5}}{1+1}=\frac{7}{5 \times 2}=\frac{7}{10}$
$\mathrm{n}=7$