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Question #42211

Difficulty: 8/10 | 🟢 Free

The Question

A solid block of volume $V$ , height $H$ , and density $\rho_b$ is suspended in air by a light elastic string of stiffness $k$ attached to a rigid support. A wide container is placed below the block and is slowly filled with a liquid of density $\rho$ (where $\rho < \rho_b$ ). As the liquid level rises, the block moves upwards due to the contraction of the string. The liquid is poured until the block is just completely submerged. What is the total work done by the buoyant force on the block during this process?

Options

A.
$\frac{\rho^2 V^2 g^2}{2k}$
B.
$\frac{\rho^2 V^2 g^2}{k}$
C.
\rho V g H
D.
$\frac{\rho^2 V^2 g^2}{2k} + \rho V g H$

Check Your Answer

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

Work Done in Stretching|Archimedes' Buoyancy Dynamics|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of work done by a variable force coupled with a holonomic constraint (the elastic string).

Students often incorrectly assume the work done by buoyancy is simply the maximum buoyant force multiplied by the block's height ( $\rho V g H$ ), or they integrate with respect to the fluid level instead of the block's absolute displacement.

The trap exploits the failure to recognize that the buoyant force is not a constant state function here, but varies linearly with the block's displacement ( $F_b = kx$ ) due to the equilibrium constraint, making the work done equal to the change in the elastic potential energy of the system.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Work Done in Stretching
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free