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Question #42824

Difficulty: 8/10 | 🟢 Free

The Question

A simple pendulum consists of a bob of mass $m$ and positive charge $q$ suspended by a massless, inextensible string of length $L$ from a fixed pivot. A fixed point charge $+Q$ is located exactly at the pivot. The entire setup is placed in a uniform horizontal electric field of magnitude $E$ . The acceleration due to gravity is $g$ downwards. The bob is allowed to reach its stable equilibrium position, where the string makes an angle $\theta_0$ with the downward vertical. If the bob is given a small angular displacement from this equilibrium position within the vertical plane, what is the time period $T$ of its small oscillations? (Assume $\epsilon_0$ is the permittivity of free space).

Options

A.
$2\pi \sqrt{\frac{L}{\left[g^2 + (qE/m)^2\right]^{1/2}}}$
B.
$2\pi \sqrt{\frac{L}{\left[g^2 + (qE/m)^2\right]^{1/2} + \frac{qQ}{4\pi\epsilon_0 m L^2}}}$
C.
$2\pi \sqrt{\frac{L}{\left[g^2 + (qE/m)^2\right]^{1/2} - \frac{qQ}{4\pi\epsilon_0 m L^2}}}$
D.
$2\pi \sqrt{\frac{L}{g + \frac{qE}{m} + \frac{qQ}{4\pi\epsilon_0 m L^2}}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Superposition Principle|Simple Pendulum Small Angle Approximation|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the student's fundamental understanding of restoring forces and effective gravity in oscillatory systems.

A common, flawed heuristic taught to students is that the effective gravity $g_{eff}$ is equal to the equilibrium tension divided by mass ( $T_{eq}/m$ ).

If applied here, students will incorrectly include the Coulomb force from the pivot in their time period calculation (Option B).

However, because the Coulomb force from the pivot is purely radial at all times, it never provides a tangential restoring component.

The true effective gravity governing the oscillation is solely the vector sum of the uniform fields (gravity and the horizontal electric field).

The trap exposes the danger of blindly applying the $T_{eq}/m$ shortcut without analyzing the tangential equation of motion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Superposition Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free