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Question #42449

Difficulty: 8/10 | 🟢 Free

The Question

An elevator accelerates downwards with a constant acceleration $a = \frac{g}{3}$ . Inside the elevator, a tall cylindrical tank filled with a viscous liquid is fixed to the floor. A small sphere of mass $m$ is released from rest at the surface of the liquid. The liquid exerts a drag force $F_d = -kv$ on the sphere, where $v$ is the velocity of the sphere relative to the liquid. Neglecting buoyant force, what is the distance fallen by the sphere relative to the elevator at the instant the apparent weight of the sphere (defined here as the magnitude of the drag force exerted by the liquid) reaches half of its maximum possible value?

Options

A.
$\frac{2m^2g}{3k^2} \left( \ln 2 - \frac{1}{2} \right)$
B.
$\frac{m^2g}{3k^2} \left( \ln 2 - \frac{1}{2} \right)$
C.
$\frac{2m^2g}{3k^2} \left( \ln 2 + \frac{1}{2} \right)$
D.
$\frac{m^2g}{k^2} \left( \ln 2 - \frac{1}{2} \right)$

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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

Apparent Weight in Accelerating Elevators|Pseudo Force Directionality|Motion in a Straight Line

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate non-inertial frame dynamics with velocity-dependent resistive forces.

A common trap is calculating the terminal velocity and maximum drag force using the ground frame gravity 'g' instead of the effective gravity 'g_eff = g - a' in the elevator frame.

Furthermore, it requires setting up and solving a non-homogeneous differential equation for position as a function of velocity, discriminating students who rely on standard kinematic formulas from those who can derive motion equations from first principles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Apparent Weight in Accelerating Elevators
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free