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Question #42316

Difficulty: 8/10 | 🟢 Free

The Question

A small spherical particle of mass $m$ and radius $r$ is released from rest in a large column of viscous fluid. The fluid has a coefficient of viscosity $\eta$ and its density is negligible compared to the density of the particle. The particle falls under the uniform gravitational field $g$ . Assuming Stokes' law is valid throughout the motion, what is the work done by the viscous drag force on the particle from the instant of release until it attains exactly half of its terminal velocity?

Options

A.
$\frac{m^3 g^2}{36 \pi^2 \eta^2 r^2} \left( \frac{5}{8} - \ln 2 \right)$
B.
$\frac{m^3 g^2}{36 \pi^2 \eta^2 r^2} \left( \ln 2 - \frac{5}{8} \right)$
C.
$\frac{m^3 g^2}{36 \pi^2 \eta^2 r^2} \left( \frac{1}{8} - \ln 2 \right)$
D.
$\frac{m^3 g^2}{72 \pi^2 \eta^2 r^2} \left( \frac{5}{8} - \ln 2 \right)$

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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

Conservative Force Field|Stokes' Law|Energy Dissipation|Differential Equation

Why This Question Is Tricky

Common Mistakes

This question targets the 'Conservative Bias' where students incorrectly apply pure mechanical energy conservation or constant acceleration kinematics ( $v^2 = u^2 + 2as$ ) to a dissipative system.

By introducing Stokes' drag, the net force becomes velocity-dependent, requiring a differential equation approach to find the path-dependent work.

Students often fail to recognize that the work done by the non-conservative viscous force must be derived via the Work-Energy Theorem after integrating the equation of motion to find the exact displacement, rather than assuming a linear energy loss or constant acceleration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Conservative Force Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free