Search any question & find its solution
Question:
Answered & Verified by Expert
A pendulum has a time period $\mathrm{T}$ in air. When it is made to oscillate in water its time period is $\sqrt{2} \mathrm{~T}$. Then the relative density of the material of the bob of the pendulum is (neglect damping)
Options:
Solution:
2977 Upvotes
Verified Answer
The correct answer is:
$2$
The effective acceleration of a bob in water
$$
g^{\prime}=g\left(1-\frac{\sigma}{\rho}\right)
$$
Here $\sigma$ and $\rho$ are the density of water and the bob respectively.
The time period in air is
$$
T=2 \pi \sqrt{\frac{I}{g}}
$$
The time period in water is
$$
T=2 \pi \sqrt{\frac{I}{g^{\prime}}}
$$
Divide equation (2) by (1), we have
$$
\begin{aligned}
& \frac{T^{\prime}}{T}=\sqrt{\frac{I}{g^{\prime}}} \times \sqrt{\frac{g}{I}} \\
& \frac{\sqrt{2} T}{T}=\sqrt{\frac{g}{g\left(1-\frac{\sigma}{\rho}\right)}} \\
& \sqrt{2}=\sqrt{\frac{1}{1-\frac{\sigma}{\rho}}}
\end{aligned}
$$
Since $(\because \sigma=1)$
$$
\begin{aligned}
& 2=\frac{1}{1-\frac{1}{\rho}} \Rightarrow 1-\frac{1}{\rho}=\frac{1}{2} \quad \Rightarrow \frac{1}{\rho}=\frac{1}{2} \\
& \therefore \rho=2
\end{aligned}
$$
$$
g^{\prime}=g\left(1-\frac{\sigma}{\rho}\right)
$$
Here $\sigma$ and $\rho$ are the density of water and the bob respectively.
The time period in air is
$$
T=2 \pi \sqrt{\frac{I}{g}}
$$
The time period in water is
$$
T=2 \pi \sqrt{\frac{I}{g^{\prime}}}
$$
Divide equation (2) by (1), we have
$$
\begin{aligned}
& \frac{T^{\prime}}{T}=\sqrt{\frac{I}{g^{\prime}}} \times \sqrt{\frac{g}{I}} \\
& \frac{\sqrt{2} T}{T}=\sqrt{\frac{g}{g\left(1-\frac{\sigma}{\rho}\right)}} \\
& \sqrt{2}=\sqrt{\frac{1}{1-\frac{\sigma}{\rho}}}
\end{aligned}
$$
Since $(\because \sigma=1)$
$$
\begin{aligned}
& 2=\frac{1}{1-\frac{1}{\rho}} \Rightarrow 1-\frac{1}{\rho}=\frac{1}{2} \quad \Rightarrow \frac{1}{\rho}=\frac{1}{2} \\
& \therefore \rho=2
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.