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Question #42325

Difficulty: 8/10 | 🟢 Free

The Question

A horizontal Venturi tube has a wide section of cross-sectional area $A$ (at point 1) and a throat of area $A/2$ (at point 2). An ideal, incompressible fluid of density $\rho$ flows through the tube. The volume flow rate is non-steady and given by $Q(t) = \beta t$ , where $\beta$ is a positive constant. A differential pressure sensor connects point 1 and point 2. The sensor consists of a small cylinder of area $a$ and a massless piston attached to a spring of stiffness $k$ . The geometric integral along the streamline from point 1 to point 2 is given as $\int_1^2 \frac{dx}{A(x)} = \frac{L}{A}$ . Assuming the volume of fluid drawn by the sensor is negligible and the piston adjusts instantaneously, what is the compression $x$ of the spring at time $t$ ?

Options

A.
$\frac{a \rho \beta}{k A} \left( \frac{3 \beta t^2}{2 A} + L \right)$
B.
$\frac{3 a \rho \beta^2 t^2}{2 k A^2}$
C.
$\frac{a \rho \beta}{k A} \left( \frac{3 \beta t^2}{2 A} - L \right)$
D.
$\frac{a \rho \beta}{k A} \left( \frac{\beta t^2}{2 A} + L \right)$

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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

Spring Potential Energy|Bernoulli’s Energy Balance|Venturi Meter Logic

Why This Question Is Tricky

Common Mistakes

This question tests the application of the unsteady Bernoulli equation in a non-steady flow regime.

Students commonly fall into the trap of reflexively applying the steady-state Bernoulli equation ( $\Delta P = \frac{1}{2}\rho(v_2^2 - v_1^2)$ ), completely ignoring the inertial term ( $\rho \int \frac{\partial v}{\partial t} ds$ ) which accounts for the local acceleration of the fluid.

By coupling this with a spring-loaded piston, the problem forces students to recognize that the pressure difference driving the mechanical sensor has both a time-dependent dynamic component and a purely inertial component, distinguishing top-tier candidates who understand the fundamental Euler equation integration from those who merely memorize the steady-state formula.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Spring Potential Energy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free