A circular disc reaches from top to bottom of an inclined plane of length \(l\). When it slips down the plane, if takes \(\mathrm{t} \mathrm{s}\). When it rolls down the plane then it takes \(\left(\frac{\alpha}{2}\right)^{1 / 2} \mathrm{t} s\), where \(\alpha\) is. _______.

\(\mathrm{a}=\mathrm{g} \sin \theta\)

\(\ell=\frac{1}{2} \mathrm{at}^2 \Rightarrow \mathrm{t}=\sqrt{\frac{2 \ell}{\mathrm{g} \sin \theta}}\)

For rolling

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

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

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

\(\Rightarrow \mathrm{t}^{\prime}=\sqrt{\frac{6 \ell}{2 \mathrm{~g} \sin \theta}}=\sqrt{\frac{\alpha}{2}} \sqrt{\frac{2 \ell}{\mathrm{s} \sin \theta}}\)

\(\Rightarrow \alpha=3 \)