Question

A pendulum has time period $T$ in air when it is made to oscillate in water, it acquired a time period $T =\sqrt{2} T$. The density of the pendulum bob is equal to (density of water $=1$ )

• A. $ \sqrt{2} $
• B. $2$
• C. $ 2\sqrt{2} $
• D. None of these

Answer 1

Answer: (B)

The effective acceleration of a bob in water
$=g=g\left[1-\frac{\sigma}{\rho}\right]$
where $a$ and $\rho$ are the densities of water and the bob respectively.
Since, the periods of oscillation of the bob in air and water are given as
$T=2 \pi \sqrt{\frac{l}{g}} $ and $ T=2 \pi \sqrt{\frac{l}{g}} $
$\therefore \frac{T}{T}=\sqrt{\frac{g}{g}}=\sqrt{\frac{g\left(1-\frac{\sigma}{\rho}\right)}{g}}$
$=\sqrt{1-\frac{\sigma}{\rho}}=\sqrt{1-\frac{1}{\rho}}$
Putting $\frac{T}{T}=\frac{1}{\sqrt{2}}$ We obtain,
$\frac{1}{2}=1-\frac{1}{\rho} $
$\Rightarrow \rho=2$