A simple pendulum oscillating in air has a period of $\sqrt{3} \mathrm{~s}$. If it is completely immersed in…
A simple pendulum oscillating in air has a period of $\sqrt{3} \mathrm{~s}$. If it is completely immersed in non-viscous liquid, having density $\left(\frac{1}{4}\right)^{\text {th }}$ of the material of the bob, the new period will be
$2 \mathrm{~s}$
$\frac{\sqrt{3}}{2} s$
$2 \sqrt{3} s$
$\frac{2}{\sqrt{3}} \mathrm{~s}$
Solution
Time period of simple pendulum
$\begin{aligned} & T=2 \pi \sqrt{\frac{l}{g_{\mathrm{eff}}}} \\ & \sqrt{3}=2 \pi \sqrt{\frac{l}{g}}\end{aligned}$
After immersed in liquid
$\begin{aligned} & M g_{\text {eff }}=M g-\frac{M g}{\frac{\rho_s}{\rho_l}} \\ & M g_{\text {eff }}=M g-\frac{M g}{4}=\frac{3 M g}{4} \\ & g_{\text {eff }}=\frac{3 g}{4} \\ & T_1=2 \pi \sqrt{\frac{l}{3 g}}=\left(2 \pi \sqrt{\frac{I}{g}}\right) \times \frac{2}{\sqrt{3}} \\ & T_1=\frac{2 T}{\sqrt{3}}=2 \mathrm{~s}\end{aligned}$