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Question #42752

Difficulty: 8/10 | 🟢 Free

The Question

A small bead of mass $m$ and positive charge $q$ is constrained to move along a smooth semi-circular track of radius $R$ . The track is fixed on a rough horizontal table with a coefficient of kinetic friction $\mu$ . The semi-circle spans from point $A(R, 0)$ to point $B(-R, 0)$ in the $xy$ -plane. Two fixed point charges, $Q_1 = -6Q$ and $Q_2 = +6Q$ (where $Q > 0$ ), are located on the table at $(2R, 0)$ and $(-2R, 0)$ respectively. What is the minimum initial kinetic energy $K_0$ that must be given to the bead at $A$ so that it just reaches $B$ ? (Assume $\epsilon_0$ is the permittivity of free space)

Options

A.
$\frac{2qQ}{\pi\epsilon_0 R} + \mu m g \pi R$
B.
$\frac{2qQ}{\pi\epsilon_0 R} + 2\mu m g R$
C.
$\frac{qQ}{\pi\epsilon_0 R} + \mu m g \pi R$
D.
$\frac{qQ}{\pi\epsilon_0 R} + 2\mu m g R$

Check Your Answer

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 of Potentials|Work Done by Friction|Work-Energy Theorem Validity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize conservative and non-conservative forces using the Work-Energy Theorem.

The primary trap lies in the cognitive dissonance between evaluating path-independent and path-dependent quantities.

Students often correctly use the superposition principle to find the change in electrostatic potential energy (which only depends on initial and final positions).

However, they frequently misapply this same scalar displacement logic to the non-conservative frictional force, incorrectly calculating the work done by friction using the straight-line displacement ( $2R$ ) instead of integrating over the actual constrained semi-circular path length ( $\pi R$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Superposition of Potentials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free