Physics - SHM Question with Solution | TestHub

For rigid body $\mathrm{T}=2\pi \sqrt{\frac{\mathrm{I}}{\mathrm{mgl}}}$

(Suppose $\ell$ is distance of point of pivoting from COM)

$=2\pi \sqrt{\frac{\left(\frac{{\mathrm{mR}}^2}{2}+\mathrm{m}{\ell }^2\right)}{\mathrm{mg}\ell }}=2\pi \sqrt{\frac{{\mathrm{R}}^2+2{\ell }^2}{2\mathrm{g}\ell }}$ $L_{\text{eff }}=\frac{R^2+2{\ell }^2}{\ell }$

should be minimum for $T$ to be minimum

$\begin{array}{rlrlr}& \frac{{\mathrm{dL}}_{\mathrm{eff}}}{\mathrm{dl}}=\frac{\ell (+4\ell )-\left({\mathrm{R}}^2+2{\ell }^2\right)}{{\ell }^2}=0& & & \end{array}$

$4{\ell }^2-R^2-2{\ell }^2=0$

$R^2=2{\ell }^2$

$\frac{R}{\sqrt{2}}=\ell$