Question

The bob of a simple pendulum executes simple harmonic motion in water with a period $t$, while the period of oscillation of the bob is $t_0$ in air. Neglecting frictional force of water and given that the density of the bob is $\left(\frac{4}{3}\right)1000kg/m^3$. What relationship between $t$ and $t_0$ is true?

• A. $t=t_0$
  
• B. $t=4t_0$
  
• C. $t=2t_0$
• D. $t=\frac{t_0}{2}$

Answer 1

The correct option is C

$t=2t_0$

When immersed in water, effective value of $g$ is $g^{\prime }=\left(\frac{\rho -{\rho }_w}{\rho }\right)g$ where $\rho$ is density of object and ${\rho }_w$ is that of water.
Now, $\frac{t}{t_0}=\frac{2\pi \sqrt{\frac{l}{g^{\prime }}}}{2\pi \sqrt{\frac{l}{g}}}=\sqrt{\frac{g}{g^{\prime }}}=\sqrt{\frac{\rho }{\rho -{\rho }_w}}$
Substituting values, we get $t=2t_0$