A pendulum is formed by pivoting a disc. What distance from center of mass, it should be pivoted for minimum time period while performing $SHM$ ?
Diffcult
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For rigid body $\mathrm{T}=2 \pi \sqrt{\frac{1}{\mathrm{mgl}}}$
(Suppose $\ell$ is distance of point of pivoting from $COM$)
$=2 \pi \sqrt{\frac{\left(\frac{m R^{2}}{2}+m \ell^{2}\right)}{m g \ell}}=2 \pi \sqrt{\frac{R^{2}+2 \ell^{2}}{2 g \ell}}$
$L_{eff}=\frac{R^{2}+2 \ell^{2}}{\ell}$ should be minimum for $T$ to be minimum
$\frac{d L_{e f f}}{d l}=\frac{\ell(+4 \ell)-\left(R^{2}+2 \ell^{2}\right)}{\ell^{2}}=0$
$4 \ell^{2}-K^{2}-2 \ell^{2}=0$
$\mathrm{R}^{2}-2 \ell^{2}$
$\frac{R}{\sqrt{2}}=\ell$
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