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Question #42223

Difficulty: 8/10 | 🟢 Free

The Question

A fixed rigid container has a cross-sectional area that decreases with height. The base of the container is a massless, frictionless piston of area $A_0$ . The container contains an ideal, incompressible fluid of density $\rho$ . The piston is pushed upwards, moving with a constant vertical velocity $v_0$ . At a certain instant, the fluid's free surface is at a height $H$ above the piston, where the container's cross-sectional area is $A_1 = \frac{A_0}{2}$ . Atmospheric pressure is $P_0$ and the acceleration due to gravity is $g$ . What is the external upward force $F$ required on the piston at this instant?

Options

A.
$P_0 A_0 + \rho A_0 g H$
B.
$P_0 A_0 + \rho A_0 g H + \frac{1}{2} \rho A_0 v_0^2$
C.
$P_0 A_0 + \rho A_0 g H + \frac{3}{2} \rho A_0 v_0^2$
D.
$P_0 A_0 + \rho A_0 g H - \frac{3}{2} \rho A_0 v_0^2$

Check Your Answer

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

Hydrostatic Paradox|Work-Energy Theorem Validity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy theorem and Bernoulli's principle in a system with a variable geometry constraint.

A common trap is the 'Hydrostatic Paradox' misconception, where students assume the force on the piston is purely static ( $F = P_0 A_0 + \rho A_0 g H$ ), ignoring the kinetic energy required to accelerate the fluid through the narrowing container.

Because the cross-section decreases, the fluid undergoes convective acceleration even though the piston's velocity is constant.

Failing to account for the spatial acceleration of the fluid or the rate of change of the fluid's kinetic energy leads to missing the dynamic pressure term $\frac{3}{2}\rho A_0 v_0^2$ .

Quick Facts About This Question

Question Information

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