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Question #42102

Difficulty: 8/10 | 🟢 Free

The Question

A large cylindrical tank of cross-sectional area $A$ contains water of density $\rho$ . A block of ice of density $\eta \rho$ (where $\eta < 1$ ) is anchored to the bottom of the tank so that it is completely submerged. A small hole of area $a$ is opened at the bottom of the tank, and water flows out. Simultaneously, a heater supplies heat at a constant rate $P$ , which is entirely used to melt the ice. The latent heat of fusion of ice is $L$ . If the instantaneous height of the water level is $h$ , what is the magnitude of the rate of fall of the water level, $\left|\frac{dh}{dt}\right|$ ?

Options

A.
$\frac{a}{A}\sqrt{2gh} + \frac{P}{AL\rho}\left(\frac{1-\eta}{\eta}\right)$
B.
$\frac{a}{A}\sqrt{2gh} - \frac{P}{AL\rho}\left(\frac{1-\eta}{\eta}\right)$
C.
$\frac{a}{A}\sqrt{2gh} + \frac{P}{AL\rho}(1-\eta)$
D.
$\frac{a}{A}\sqrt{2gh}$

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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

Torricelli’s Efflux Velocity|Latent Heat and Volume Change|Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of fluid dynamics (Torricelli's Law) with thermodynamics (phase transitions).

The primary trap is assuming that the water level drops solely due to the efflux of water from the hole, leading students to incorrectly choose the standard continuity result (Option D).

However, because the ice is fully submerged (anchored), its melting causes a continuous volume contraction of the system since the density of water is greater than that of ice.

Students must correctly formulate the differential equation for volume balance, recognizing that the phase change introduces a time-dependent negative volume term proportional to the melting rate.

A secondary trap involves incorrectly calculating the specific volume difference, leading to Option C.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Torricelli’s Efflux Velocity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free