A lift is ascending with acceleration $\frac{g}{3}.$ Find out the time period of a simple pendulum suspended from its ceiling if its time period in the stationary lift is T.

Time period of a simple pendulum can be given in a modified form as $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{{\mathrm{g}}_{\mathrm{cft}}}},$ where g_eff is the effective value of restoring acceleration.

When lift is ascending with $\frac{g}{3},$ then ${\mathrm{g}}_{\mathrm{eff}}=\mathrm{g}+\frac{\mathrm{g}}{3}=\frac{4\mathrm{g}}{3}.$ 

Hence, time period in this case will be

$\begin{array}{l}{\mathrm{T}}^{\mathrm{\prime }}=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{{\mathrm{g}}_{\mathrm{eff}}}}=2\mathrm{\pi }\sqrt{\frac{3\mathrm{l}}{4\mathrm{g}}}=\frac{\sqrt{3}}{2}\mathrm{T}\\ \left(\text{ Here },\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}\right)\end{array}$