The following bodies, (1) a ring (2) a disc (3) a solid cylinder (4) a solid sphere of same mass 'm' and radius 'R' are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is _______. [Mark the body as per their respective numbering given in the question] Option: 1 4 Option: 2 3 Option: 3 2 Option: 4 1

$\begin{array}{l} \tau =I\alpha \\ \mathrm{Mg} \sin \theta \mathrm{R}=\left(\mathrm{mk}^{2}+\mathrm{mR}^{2}\right) \alpha \\ \\ \alpha=\frac{\mathrm{Rg} \sin \theta}{\mathrm{k}^{2}+\mathrm{R}^{2}} \\ \\ \Rightarrow \mathrm{a}=\frac{\mathrm{g} \sin \theta}{1+\frac{\mathrm{k}^{2}}{\mathrm{R}^{2}}} \end{array}$

$\mathrm{t}=\sqrt{\frac{2 \mathrm{~s}}{\mathrm{a}}}=\sqrt{\frac{2 \mathrm{~s}}{\mathrm{~g} \sin \theta}\left(1+\frac{\mathrm{k}^{2}}{\mathrm{R}^{2}}\right)}$

for least time, k should be the least

and k is least for the solid sphere.

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