JEE Main 2024 — 9 April, Shift 2 — Question 50

For slipping

$\mathrm{a}=\mathrm{g} \sin \theta$

$\ell=\frac{1}{2} a t^{2} \Rightarrow t=\sqrt{\frac{2 \ell}{g \sin \theta}}$

For rolling

$\mathrm{a}^{\prime}=\frac{\mathrm{g} \sin \theta}{1+\frac{\mathrm{k}^{2}}{\mathrm{R}^{2}}}\left[\mathrm{k}=\frac{\mathrm{R}}{\sqrt{2}}\right]$

$\Rightarrow \mathrm{a}^{\prime}=\frac{2 \mathrm{~g} \sin \theta}{3}$

$\ell=\frac{1}{2} \mathrm{a}^{\prime}\left(\mathrm{t}^{\prime}\right)^{2}$

$\Rightarrow \mathrm{t}^{\prime}=\sqrt{\frac{6 \ell}{2 g \sin \theta}}=\sqrt{\frac{\alpha}{2}} \sqrt{\frac{2 \ell}{g \sin \theta}}$

$\Rightarrow \alpha=3$