Q 11-06-080JEE MainJEE Main 2024 (4 Apr, Shift 1)Medium
A solid sphere and a hollow cylinder roll up without slipping on the same inclined plane with the same initial speed $v$. The sphere and the cylinder reach up to maximum heights $h_1$ and $h_2$, respectively, above the initial level. The ratio $h_1 : h_2$ is $\dfrac{n}{10}$. The value of $n$ is ______.
Numerical value type. Enter your answer.
Answer: 7
For rolling, $\dfrac12mv^2\left(1 + \dfrac{k^2}{R^2}\right) = mgh$.
Solid sphere: $\dfrac{k^2}{R^2} = \dfrac25 \Rightarrow h_1 = \dfrac{7v^2}{10g}$.
Hollow cylinder: $\dfrac{k^2}{R^2} = 1 \Rightarrow h_2 = \dfrac{v^2}{g}$.
$$\frac{h_1}{h_2} = \frac{7}{10} \Rightarrow n = 7$$
Solution by Sreeraj P, M.Sc Physics