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6.System of Particles and Rotational Motion
hard
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__________.
A
$6$
B
$7$
C
$8$
D
$9$
(JEE MAIN-2024)
Solution
$\text { Gain in P.E. }=\text { Loss in K.E. }$
$\mathrm{mgh}=\frac{1}{2} \mathrm{mv}^2\left(1+\frac{\mathrm{K}^2}{\mathrm{R}^2}\right)$
$\mathrm{h} \propto 1+\frac{\mathrm{K}^2}{\mathrm{R}^2}$
$\frac{\mathrm{h}_1}{\mathrm{~h}_2}=\frac{1+\frac{2}{5}}{1+1}=\frac{7}{5 \times 2}=\frac{7}{10}$
$\mathrm{n}=7$
Standard 11
Physics
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