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Question #43908

Difficulty: 8/10 | 🟢 Free

The Question

A U-tube consists of two vertical capillary arms of radii $r$ and $2r$ , connected at their bottoms by a short horizontal tube of negligible volume. The U-tube is filled with a liquid of density $\rho$ and surface tension $\gamma$ exactly up to the brims of both arms. The contact lines of the liquid are pinned at the top edges of the arms, so the menisci are initially perfectly flat. The liquid is given a small volume displacement such that it bulges out at the top of one arm and bulges in at the other, forming spherical caps. Assuming the length of the liquid column in each arm is $L$ , and neglecting gravitational potential energy changes and viscous effects, what is the angular frequency $\omega$ of the resulting small oscillations?

Options

A.
$\sqrt{\frac{34 \gamma}{5 \rho L r^2}}$
B.
$\sqrt{\frac{32 \gamma}{5 \rho L r^2}}$
C.
$\sqrt{\frac{17 \gamma}{10 \rho L r^2}}$
D.
$\sqrt{\frac{17 \gamma}{5 \rho L r^2}}$

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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

Capillary Rise Mechanics|Effective Spring Constant|Work-Energy Theorem

Why This Question Is Tricky

Common Mistakes

This question tests two advanced concepts in oscillatory dynamics and fluid mechanics.

First, students must recognize that when a liquid is pinned at the edges, the restoring force arises purely from the change in surface area (surface tension), acting as an effective spring.

A common mistake is miscalculating the volume of the spherical cap ( $V \approx \frac{\pi}{2} r^2 h$ instead of $\pi r^2 h$ ), leading to an incorrect relation between displacement and cap height.

Second, the unequal bores create an asymmetry in the velocity distribution.

Students often fall into the trap of using the simple total mass ( $5\rho \pi r^2 L$ ) instead of calculating the true effective mass ( $M_{eff}$ ) via kinetic energy equivalence, which accounts for the different flow velocities in each arm.

Quick Facts About This Question

Question Information

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