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Question #42893

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The Question

A simple pendulum consists of a small bob of mass $m$ and positive charge $q$ suspended by a massless, inextensible string of length $L$ . The point of suspension is at $O$ . The equilibrium position of the bob is at $P$ , where $OP = L$ . A fixed point charge $+Q$ is placed at point $S$ , such that $P$ is the midpoint of the vertical line segment $OS$ (with $S$ located directly below $P$ ). The pendulum is displaced by a small angle $\theta$ from its equilibrium position and released. Assuming $mg > \frac{Qq}{2\pi\epsilon_0 L^2}$ , what is the angular frequency $\omega$ of the resulting simple harmonic motion?

Options

A.
$\sqrt{\frac{g}{L} - \frac{Qq}{2\pi\epsilon_0 m L^3}}$
B.
$\sqrt{\frac{g}{L} - \frac{Qq}{4\pi\epsilon_0 m L^3}}$
C.
$\sqrt{\frac{g}{L} + \frac{Qq}{4\pi\epsilon_0 m L^3}}$
D.
$\sqrt{\frac{g}{L} - \frac{Qq}{8\pi\epsilon_0 m L^3}}$

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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

Coulomb's Law Vector Form|Gravitational Potential Energy (System)|Simple Pendulum Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of small-angle approximations in coupled force fields.

A common trap is to assume the electrostatic force acts purely vertically with a constant magnitude $F_e = \frac{Qq}{4\pi\epsilon_0 L^2}$ at all small angles, leading to an incorrect effective gravity $g_{eff} = g - \frac{F_e}{m}$ (Option B).

However, as the bob moves, the distance vector $\vec{r}$ changes both direction and magnitude.

The tangential restoring force receives equal linear contributions from both the $x$ -component of the Coulomb force and the projection of its $y$ -component onto the tangential axis, introducing a critical factor of 2.

Students must either perform a proper Taylor expansion of the potential energy or rigorously linearize the force vector to avoid this trap.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Coulomb's Law Vector Form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free