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Question #42602

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The Question

A closed cylindrical container of total mass $M$ (including the liquid inside) is being pulled vertically upwards by a constant external force $F$ , such that it moves with an initial upward acceleration $a_0$ . The container is filled with an ideal incompressible liquid of density $\rho$ up to a height $h$ . At time $t=0$ , a small plug of cross-sectional area $s$ ( $s \ll$ container's base area) at the bottom of the container is suddenly removed, allowing the liquid to effuse downwards. Assume ideal effusion (coefficient of contraction $C_c = 1$ ). Let $m_0 = \rho s h$ be the mass of the liquid column directly above the orifice. Which of the following expressions gives the instantaneous upward acceleration $a$ of the container immediately after the plug is removed?

Options

A.
$\frac{M a_0 + 2 m_0 g}{M - 2 m_0}$
B.
$\frac{M a_0 + m_0 g}{M - m_0}$
C.
$\frac{M a_0 - 2 m_0 g}{M + 2 m_0}$
D.
$\frac{M a_0 + 2 m_0 (g + a_0)}{M}$

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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

Effusion and Diffusion|Variable Mass Systems|Accelerated Frame Hydrostatics

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of variable mass dynamics with fluid hydrostatics in a non-inertial frame.

A common trap is assuming the thrust force depends on the initial acceleration $a_0$ (Option D) or standard gravity $g$ , failing to recognize that the efflux velocity $v_e$ is instantaneously determined by the new effective gravity $g_{eff} = g + a$ , creating a coupled differential relationship.

Another major trap is incorrectly calculating the thrust force as just the static pressure difference $P \cdot s = m_0(g+a)$ (Option B), neglecting the dynamic momentum flux which introduces a factor of 2 ( $F_{th} = \rho s v_e^2 = 2P_{gauge}s$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Effusion and Diffusion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free