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Question #42925

Difficulty: 8/10 | 🟢 Free

The Question

A small particle of mass $m$ and positive charge $q$ is suspended from a fixed point $O$ by a massless, inextensible string of length $R$ . A fixed point charge $+Q$ is rigidly placed at a distance $2R$ vertically below $O$ . The particle is given a horizontal velocity $u$ at its lowest position. What is the minimum value of $u$ required for the particle to complete a full vertical circle? (Assume $mg > \frac{4kQq}{R^2}$ to ensure the string does not go slack before the highest point, and let $k = \frac{1}{4\pi\epsilon_0}$ )

Options

A.
$\sqrt{5gR - \frac{13kQq}{9mR}}$
B.
$\sqrt{5gR - \frac{kQq}{9mR}}$
C.
$\sqrt{5gR - \frac{4kQq}{3mR}}$
D.
$\sqrt{5gR + \frac{11kQq}{9mR}}$

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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

Charged Conducting Sphere|Vertical Circular Motion|Potential Energy Reference|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the superposition of gravitational and electrostatic fields in vertical circular motion.

The primary trap is failing to account for the varying electrostatic potential energy while simultaneously modifying the centripetal force equation at the highest point.

Students often mistakenly assume the electrostatic force is constant or ignore the potential energy difference (Trap B), or they account for the potential energy but forget that the electrostatic force alters the required centripetal velocity at the top (Trap C).

Another common error is a sign mistake in the potential energy difference, leading to Trap D.

Quick Facts About This Question

Question Information

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