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Question #42118

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The Question

A solid sphere of radius $r$ and uniform density $\rho$ is released from rest at the surface ( $y=0$ ) of a liquid column. The liquid has a depth-dependent density $\sigma(y) = \sigma_0(1 + \beta y)$ (where $\rho > \sigma_0$ ) and a depth-dependent viscosity $\eta(y) = \eta_0(1 + \alpha y)$ . The sphere is observed to have a velocity $v$ at a depth $y=H$ . Which of the following expressions gives the magnitude of the work done by the viscous drag force on the sphere during its descent to depth $H$ ?

Options

A.
$\frac{4}{3}\pi r^3 \left[ gH(\rho - \sigma_0) - \frac{1}{2}g\sigma_0\beta H^2 - \frac{1}{2}\rho v^2 \right]$
B.
$\frac{4}{3}\pi r^3 \left[ gH(\rho - \sigma_0) + \frac{1}{2}g\sigma_0\beta H^2 - \frac{1}{2}\rho v^2 \right]$
C.
$6\pi\eta_0 r v H \left( 1 + \frac{\alpha H}{2} \right)$
D.
$\frac{4}{3}\pi r^3 \left[ gH(\rho - \sigma_0) - \frac{1}{2}\rho v^2 \right] - 3\pi\eta_0 r v H^2 \alpha$

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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

Viscosity and Stokes' Law|Work-Energy Theorem Validity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize when to apply the Work-Energy Theorem instead of attempting to solve an intractable differential equation.

The primary trap is the depth-dependent viscosity $\eta(y)$ , which tempts students to integrate the Stokes' drag force directly as $\int 6\pi\eta(y)rv \, dy$ .

However, since the velocity profile $v(y)$ is unknown and terminal velocity is never reached, this integral cannot be evaluated directly.

By applying the Work-Energy Theorem, the non-conservative work done by the drag force can be found purely through the changes in kinetic energy and the work done by the conservative (gravity) and position-dependent (buoyancy) forces, rendering the complex viscosity function a deliberate distractor.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Viscosity and Stokes' Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free