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Question #42116

Difficulty: 8/10 | 🟢 Free

The Question

A vessel consisting of two large parallel vertical plates separated by a distance $2r$ contains a liquid of density $\rho$ and surface tension $T$ . The contact angle of the liquid with the plates is exactly $90^\circ$ . At $t=0$ , the vessel is subjected to a horizontal jerk $j$ (so its acceleration is $a(t) = jt$ ). Assuming the liquid surface adjusts quasi-statically and its slope remains small enough that the surface curvature is well-approximated by $\frac{d^2y}{dx^2}$ , find the magnitude of the difference in liquid levels between the two plates at time $t$ .

Options

A.
$\frac{2 j t}{g} \left( r - \sqrt{\frac{T}{\rho g}} \tanh\left( r \sqrt{\frac{\rho g}{T}} \right) \right)$
B.
$\frac{2 j t}{g} \left( r + \sqrt{\frac{T}{\rho g}} \tanh\left( r \sqrt{\frac{\rho g}{T}} \right) \right)$
C.
$\frac{2 j t r}{g}$
D.
$\frac{2 j t}{g} \sqrt{\frac{T}{\rho g}} \tanh\left( r \sqrt{\frac{\rho g}{T}} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Contact Angle Interpretation|Non-Uniform Acceleration Constraint|Free Body Diagram Isolation

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between non-inertial system dynamics and surface tension constraints.

A common trap is assuming that in an accelerating frame, the liquid surface simply tilts to become perpendicular to the effective gravity vector, yielding a planar surface with slope $\tan \alpha = a/g$ (leading to Option C).

However, this violates the boundary condition imposed by the contact angle.

The surface must curve to satisfy both the hydrostatic pressure gradient and the $90^\circ$ contact angle at the walls.

Surface tension acts to suppress the sloshing, requiring the setup of a differential equation (Young-Laplace) to find the true non-linear deformation.

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