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Question #42936

Difficulty: 8/10 | 🟢 Free

The Question

A small insulating sphere of mass $m$ and positive charge $+q$ is suspended in a non-conducting, incompressible liquid of density $\rho_l$ inside an elevator. The density of the sphere is $\rho_s$ ( $\rho_s > \rho_l$ ). A fixed point charge $+Q$ is located at the bottom of the liquid tank. Initially, the elevator is at rest, and the sphere is in stable equilibrium at a height $h_0$ above $+Q$ . The elevator suddenly begins to accelerate upwards with a constant acceleration $a$ ( $a \ll g$ ). Assuming the liquid tank is sufficiently large and neglecting viscous drag, what is the maximum kinetic energy of the sphere relative to the elevator during its subsequent motion? (Acceleration due to gravity is $g$ )

Options

A.
$\frac{m a^2 h_0}{4g} \left( 1 - \frac{\rho_l}{\rho_s} \right)$
B.
$\frac{m a^2 h_0}{2g} \left( 1 - \frac{\rho_l}{\rho_s} \right)$
C.
$\frac{m a^2 h_0}{4g} \left( 1 + \frac{\rho_l}{\rho_s} \right)$
D.
$\frac{m a^2 h_0}{4g}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Neutral Equilibrium Points|Accelerated Frame Hydrostatics|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in an accelerated non-inertial frame, specifically coupling it with hydrostatics and electrostatics.

The primary trap is failing to recognize that while the electrostatic force is frame-invariant, the buoyancy force is frame-dependent because the hydrostatic pressure gradient becomes $\nabla P = \rho_l(g+a)$ .

Students often incorrectly apply the pseudo force only to the particle's mass, ignoring the altered displaced fluid weight.

Furthermore, calculating the maximum kinetic energy requires recognizing that the restoring force is non-constant, necessitating an energy conservation approach and a second-order Binomial Approximation to avoid math bloat and capture the $O(a^2)$ energy term.

Quick Facts About This Question

Question Information

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