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Question #42238

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical container filled with a fluid rotates with a constant angular velocity $\omega$ about its vertical axis. The $z$ -axis is taken vertically downwards along the axis of rotation, and $r$ is the radial distance from the axis. Due to a specific thermal and solute gradient, the density of the fluid in the steady rotating frame is maintained as $\rho(r,z) = \rho_0 (1 + \alpha z + \beta r^2)$ , where $\alpha$ and $\beta$ are positive constants. The acceleration due to gravity is $g$ downwards. For this steady-state relative equilibrium to be physically possible, which of the following conditions must hold?

Options

A.
$\alpha \omega^2 = 2 \beta g$
B.
$\alpha \omega^2 = \beta g$
C.
$2 \alpha \omega^2 = \beta g$
D.
$\alpha g = 2 \beta \omega^2$

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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

Accelerated Frame Hydrostatics|Variation of Density with Temperature|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the integrability condition of the pressure field in a non-inertial reference frame with a variable density gradient.

Students often mistakenly assume that any arbitrary density distribution can be maintained in a rotating fluid.

However, because the effective gravity field is non-uniform, the pressure gradient $dP = \rho \omega^2 r dr + \rho g dz$ must be an exact differential to prevent continuous convective currents (baroclinic torque).

This requires the curl of the force field per unit volume to be zero, forcing a strict mathematical coupling between the radial and vertical density gradients.

Quick Facts About This Question

Question Information

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