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Question #77070

Difficulty: 8/10 | 🟢 Free

The Question

A large open vessel containing an ideal liquid of density $\rho$ accelerates horizontally with a constant acceleration $a = \sqrt{3}g$ . A pressure gauge is attached to a small orifice on the wall of the vessel. The gauge was calibrated in a static frame to display the gauge pressure as a liquid head $h_{\text{read}}$ (i.e., it assumes $P_{\text{gauge}} = \rho g h_{\text{read}}$ ). The gauge has a positive zero error $h_0$ (it reads $h_0$ when exposed to atmospheric pressure). Let $z$ be the perpendicular distance from the orifice to the inclined free surface, $Y$ be the vertical depth of the orifice strictly below the free surface, and $X$ be the horizontal distance from the orifice to the free surface at the same elevation. Let $v$ be the efflux velocity of the liquid relative to the vessel when the orifice is opened. Which of the following statements is INCORRECT?

Options

A.
The vertical depth of the orifice below the free surface is $Y = h_{\text{read}} - h_0$ .
B.
The perpendicular distance from the orifice to the free surface is $z = \frac{1}{2}(h_{\text{read}} - h_0)$ .
C.
The efflux velocity of the liquid relative to the vessel is $v = 2\sqrt{g(h_{\text{read}} - h_0)}$ .
D.
The horizontal distance from the orifice to the free surface at the same elevation is $X = \frac{h_{\text{read}} - h_0}{\sqrt{3}}$ .

Check Your Answer

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

Zero Error Correction|Accelerated Frame Hydrostatics|Torricelli’s Efflux Velocity|Vector Approximations

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to navigate fluid statics and dynamics in a non-inertial frame while accounting for instrumental zero error.

The primary trap lies in Option C, where students often misapply Torricelli's law by mixing the effective gravity $g_{\text{eff}}$ with the vertical depth $Y$ (calculating $v = \sqrt{2 g_{\text{eff}} Y}$ ), leading exactly to the incorrect factor of 2 outside the square root.

Furthermore, the problem stresses 3D visualization and vector calculus concepts by requiring the student to map the scalar pressure reading to vertical, horizontal, and perpendicular geometric distances ( $Y$ , $X$ , $z$ ) using the respective components of the pressure gradient ( $\rho g$ , $\rho a$ , $\rho g_{\text{eff}}$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Zero Error Correction in Instruments
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free