Physics - SHM Question with Solution | TestHub

As retardation $=\mathrm{bv}$ $\therefore$ retarding force $=\mathrm{mbv}$ $\therefore$ net restoring torque when angular displacement is $\theta$ is given by $=-\mathrm{mg}\ell \sin \theta +\mathrm{mbv}\ell$ $\therefore \mathrm{I}\alpha =-mg\ell \sin \theta +\mathrm{mbv}\ell$ where, $\mathrm{I}=\mathrm{m}{\ell }^2$ $\therefore \frac{{\mathrm{d}}^2\theta }{{\mathrm{dt}}^2}=\alpha =-\frac{\mathrm{g}}{\ell }\sin \theta +\frac{\mathrm{bv}}{\ell }$ for small damping, the solution of the above differential equation will be $\therefore \theta ={\theta }_0{\mathrm{e}}^{-\frac{\mathrm{bt}}{2}}\sin (wt+\varphi )$ $\therefore$ angular amplitude will be $=\theta .e^{\frac{-bt}{2}}$ According to question, in $\tau$ time (average life-time), angular amplitude drops to $\frac{1}{e}$ value of its original value $(\theta )$ $\therefore \frac{{\theta }_0}{\mathrm{e}}={\theta }_0{\mathrm{e}}^{-\frac{6\tau }{2}}$ $\frac{6\tau }{2}=1$ $\therefore \tau =\frac{2}{b}$