Question

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

• A. $T\frac{\rho -1}{\rho }$
• B. $T\frac{\rho }{\rho -1}$
• C. $T\sqrt{\frac{\rho -1}{\rho }}$
• D. $T\sqrt{\frac{\rho }{\rho -1}}$

Answer 1

The correct option is D $T\sqrt{\frac{\rho }{\rho -1}}$

When the bob is immersed in water its effective weight
$=\left(mg-\frac{m}{\rho }g\right)=mg\left(\frac{\rho -1}{\rho }\right)\therefore g_{eff}=g\left(\frac{\rho -1}{\rho }\right)\frac{T^{\prime }}{T}=\sqrt{\frac{g}{g_{eff}}}\Rightarrow T^{\prime }=T\sqrt{\frac{\rho }{\rho -1}}$