A solid sphere and a hollow cylinder roll up without slipping on … - Vidyadip

Question

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 $\mathrm{n}$ is__________.

✓

Answer

$\text { Gain in P.E. }=\text { Loss in K.E. }$

$\mathrm{mgh}=\frac{1}{2} \mathrm{mv}^2\left(1+\frac{\mathrm{K}^2}{\mathrm{R}^2}\right)$

$\mathrm{h} \propto 1+\frac{\mathrm{K}^2}{\mathrm{R}^2}$

$\frac{\mathrm{h}_1}{\mathrm{~h}_2}=\frac{1+\frac{2}{5}}{1+1}=\frac{7}{5 \times 2}=\frac{7}{10}$

$\mathrm{n}=7$

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