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Question #42136

Difficulty: 8/10 | 🟢 Free

The Question

A solid sphere of density $\rho_1$ at temperature $T_0$ is released from rest at the surface ( $y=0$ ) of a deep, non-viscous liquid column. The temperature of the liquid varies with depth $y$ as $T(y) = T_0 - ky$ , where $k$ is a positive constant. The density of the liquid at $T_0$ is $\rho_2$ , and it is given that $\rho_1 = \frac{5}{4}\rho_2$ . The volume expansion coefficient of the liquid is $\gamma_l = 5\alpha$ and that of the sphere's material is $\gamma_s = 2\alpha$ . Assuming $\alpha k y \ll 1$ so that higher-order terms can be neglected, what is the maximum depth $H$ reached by the sphere before it momentarily comes to rest?

Options

A.
$\frac{1}{6\alpha k}$
B.
$\frac{1}{12\alpha k}$
C.
$\frac{1}{10\alpha k}$
D.
$\frac{1}{20\alpha k}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Variation of Density with Temperature|Mechanical Energy Conservation|Archimedes Principle

Why This Question Is Tricky

Common Mistakes

This question tests the dynamic coupling of Archimedes' principle with thermal expansion and mechanical energy conservation.

The primary trap is neglecting the thermal contraction of the sphere's volume, which alters the buoyant force alongside the fluid's density gradient.

A secondary, highly lethal trap is confusing the equilibrium position (where net force is zero, yielding $1/(12\alpha k)$ ) with the maximum depth (where velocity is zero), failing to apply the Work-Energy Theorem to a non-constant force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Variation of Density with Temperature
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free