A simple pendulum oscillating in air has a period of 3 ~s . If it...

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

Options

A $2 \mathrm{~s}$
B $\frac{\sqrt{3}}{2} s$
C $2 \sqrt{3} s$
D $\frac{2}{\sqrt{3}} \mathrm{~s}$

Correct answer

A. $2 \mathrm{~s}$

Step-by-step 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}

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