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Question #42115

Difficulty: 8/10 | 🟢 Free

The Question

A large open cylindrical tank is filled with an ideal liquid of density $\rho$ and surface tension $\gamma$ . A narrow horizontal capillary tube of radius $r$ is attached to the side of the tank at a depth $h$ below the free surface of the liquid. The liquid flows out steadily through the capillary into the atmosphere, where the atmospheric pressure is $P_0$ . Assuming the liquid forms a spherical meniscus at the exit of the capillary characterized by a contact angle $\theta$ , what is the steady efflux velocity $v$ of the liquid? (Assume $r$ is small enough that gravitational distortion of the meniscus is negligible).

Options

A.
$\sqrt{2gh + \frac{4\gamma \cos\theta}{\rho r}}$
B.
$\sqrt{2gh - \frac{4\gamma \cos\theta}{\rho r}}$
C.
$\sqrt{2gh + \frac{2\gamma \cos\theta}{\rho r}}$
D.
$\sqrt{2gh - \frac{2\gamma \cos\theta}{\rho r}}$

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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

Contact Angle Interpretation|Torricelli’s Efflux Velocity|Bernoulli’s Energy Balance

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of fluid dynamics (Bernoulli's principle) and surface tension (Young-Laplace pressure).

The primary trap is the blind application of Torricelli's law ( $v = \sqrt{2gh}$ ), ignoring the pressure jump across the curved meniscus at the capillary orifice.

Students must recognize that the contact angle dictates the meniscus curvature, which introduces an additional effective pressure head.

A common mistake is either omitting this term entirely or applying the incorrect sign convention for the capillary pressure (e.g., confusing the pressure inside a cylindrical jet with the pressure behind a spherical meniscus), leading to an erroneous velocity expression.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Contact Angle Interpretation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free