Simple Harmonic Motion 5 Question 1
1. A simple pendulum oscillating in air has period $T$. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $\frac{1}{16}$ th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is
(a) $2 T \sqrt{\frac{1}{10}}$
(b) $2 T \sqrt{\frac{1}{14}}$
(c) $4 T \sqrt{\frac{1}{14}}$
(d) $4 T \sqrt{\frac{1}{15}}$
(Main 2019, 9 April I)
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Answer:
Correct Answer: 1. (d)
Solution:
- We know that,
Time period of a pendulum is given by
$$ T=2 \pi \sqrt{L / g _{\text {eff }}} $$
Here, $L$ is the length of the pendulum and $g _{\text {eff }}$ is the effective acceleration due to gravity in the respective medium in which bob is oscillating.
Initially, when bob is oscillating in air, $g _{\text {eff }}=g$.
So, initial time period, $T=2 \pi \sqrt{\frac{L}{g}}$
Let $\rho _{\text {bob }}$ be the density of the bob.
When this bob is dipped into a liquid whose density is given as
$$ \rho _{\text {liquid }}=\frac{\rho _{\text {bob }}}{16}=\frac{\rho}{16} $$
$\therefore$ Net force on the bob is
$$ F _{\text {net }}=V \rho g-V \cdot \frac{\rho}{16} \cdot g $$
(where, $V=$ volume of the bob $=$ volume of displaced liquid by the bob when immersed in it). If effective value of gravitational acceleration on the bob in this liquid is $g _{\text {eff }}$, then net force on the bob can also be written as
$$ F _{\text {net }}=V \rho g _{\text {eff }} $$
Equating Eqs. (iii) and (iv), we have
$$ \begin{aligned} \Rightarrow & V \rho g _{\text {eff }} & =V \rho g-V \rho g / 16 \\ & g _{\text {eff }} & =g-g / 16=\frac{15}{16} g \end{aligned} $$
Substituting the value of $g _{\text {eff }}$ from Eq. (v) in Eq. (i), the new time period of the bob will be
$$ \begin{aligned} T^{\prime} & =2 \pi \sqrt{\frac{L}{g _{\text {eff }}}}=2 \pi \sqrt{\frac{16}{15} \frac{L}{g}} \Rightarrow T^{\prime}=\sqrt{\frac{16}{15}} \times 2 \pi \sqrt{\frac{L}{g}} \\ & =\frac{4}{\sqrt{15}} \times T \quad \text { [using Eq. (ii)] } \end{aligned} $$
