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(D) When the bob is immersed in water,the effective weight $W_{eff}$ is given by the actual weight minus the buoyant force.$W_{eff} = mg - F_B = mg -

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$T\sqrt{\frac{\rho}{\rho - 1}}$

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(D) When the bob is immersed in water,the effective weight $W_{eff}$ is given by the actual weight minus the buoyant force.$W_{eff} = mg - F_B = mg - V\rho_w g$Since the relative density $\rho = \frac{\rho_{bob}}{\rho_w}$,we have $V = \frac{m}{\rho_{bob}} = \frac{m}{\rho \rho_w}$.$W_{eff} = mg - \left(\frac{m}{\rho \rho_w}\right) \rho_w g = mg \left(1 - \frac{1}{\rho}\right) = mg \left(\frac{\rho - 1}{\rho}\right)$.Thus,the effective acceleration due to gravity is $g_{eff} = g \left(\frac{\rho - 1}{\rho}\right)$.The time period of a simple pendulum is $T = 2\pi \sqrt{\frac{L}{g}}$.Therefore,the new time period $T'$ is $T' = 2\pi \sqrt{\frac{L}{g_{eff}}} = 2\pi \sqrt{\frac{L}{g \left(\frac{\rho - 1}{\rho}\right)}} = T \sqrt{\frac{\rho}{\rho - 1}}$.

The metallic bob of a simple pendulum has the relative density $\rho$. The time period of this pendulum is $T$. If the metallic bob is immersed in water,then the new time period is given by

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