SATHEE: Simple Harmonic Motion 5 Question 1

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) $2 T \sqrt{\frac{1}{14}}$

(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

$T = 2 π \sqrt{L / g_{eff}}$

Here, $L$ is the length of the pendulum and $g_{eff}$ is the effective acceleration due to gravity in the respective medium in which bob is oscillating.

Initially, when bob is oscillating in air, $g_{eff} = g$ .

So, initial time period, $T = 2 π \sqrt{\frac{L}{g}}$

Let $ρ_{bob}$ be the density of the bob.

When this bob is dipped into a liquid whose density is given as

$ρ_{liquid} = \frac{ρ_{bob}}{16} = \frac{ρ}{16}$

$∴$ Net force on the bob is

$F_{net} = V ρ g - V \cdot \frac{ρ}{16} \cdot g$

(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 $g_{eff}$ , then net force on the bob can also be written as

$F_{net} = V ρ g_{eff}$

Equating Eqs. (iii) and (iv), we have

$\begin{aligned} \Rightarrow & V ρ g_{eff} & = V ρ g - V ρ g / 16 \\ g_{eff} & = g - g / 16 = \frac{15}{16} g \end{aligned}$

Substituting the value of $g_{eff}$ from Eq. (v) in Eq. (i), the new time period of the bob will be

$\begin{aligned} T^{'} & = 2 π \sqrt{\frac{L}{g_{eff}}} = 2 π \sqrt{\frac{16}{15} \frac{L}{g}} \Rightarrow T^{'} = \sqrt{\frac{16}{15}} \times 2 π \sqrt{\frac{L}{g}} \\ = \frac{4}{\sqrt{15}} \times T [using Eq. (ii)] \end{aligned}$

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

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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 116 th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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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