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Question #42622

Difficulty: 8/10 | 🟢 Free

The Question

A simple pendulum consists of a bob of mass $m$ , density $\rho$ , and charge $q$ , suspended by a light string of length $L$ . The pendulum is located in a mine at a depth $h = R/2$ below the Earth's surface (where $R$ is the radius of the Earth and $g$ is the surface gravity). The pendulum oscillates between two large, isolated, vertical parallel plates carrying charges $+Q$ and $-Q$ . In this setup, the pendulum's mean position makes an angle $\theta_1$ with the vertical, where $\tan \theta_1 = 4/3$ . A non-viscous liquid dielectric of dielectric constant $K = 5/3$ and density $\rho_L = \rho/5$ is then poured into the space between the plates, completely submerging the pendulum. Ignoring viscous drag and hydrodynamic added-mass effects, what is the new time period $T_2$ of small oscillations of the pendulum?

Options

A.
$2\pi \sqrt{\frac{5L}{2\sqrt{2}g}}$
B.
$2\pi \sqrt{\frac{5L}{4\sqrt{2}g}}$
C.
$2\pi \sqrt{\frac{10L}{\sqrt{41}g}}$
D.
$2\pi \sqrt{\frac{15L}{2\sqrt{34}g}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Simple Pendulum Small Angle Approximation|Gravitational Field Variation|Dielectric Slab Displacement

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of gravitation, electrostatics in dielectrics, and fluid mechanics.

The primary trap lies in assuming the electric field remains constant when the dielectric is inserted; for an isolated capacitor, the charge remains constant, so the electric field and the resulting electrostatic force decrease by a factor of $K$ .

Additionally, students often forget to account for the variation of gravity with depth ( $g' = g/2$ ) or neglect the buoyant force introduced by the liquid dielectric, both of which are required to correctly determine the new effective restoring acceleration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Simple Pendulum Small Angle Approximation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free