Before You Begin

Question #42221

Difficulty: 8/10 | 🟢 Free

The Question

A closed conical container of height $H$ and base area $A$ is placed with its flat base on a horizontal surface and its vertex pointing vertically upwards. The container is completely filled with a non-uniform incompressible fluid whose density varies with depth $y$ (measured downwards from the vertex) as $\rho(y) = \rho_0 \left(1 + \frac{y}{H}\right)$ , where $\rho_0$ is a positive constant. Let $g$ be the acceleration due to gravity. Neglecting atmospheric pressure, what is the magnitude of the net vertical force exerted by the curved inner walls of the container on the fluid?

Options

A.
$\frac{11}{12} \rho_0 g A H$
B.
$\frac{7}{12} \rho_0 g A H$
C.
$\frac{3}{2} \rho_0 g A H$
D.
$\frac{2}{3} \rho_0 g A H$

Check Your Answer

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

Hydrostatic Paradox|Potential vs Field Relationship|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the hydrostatic paradox and fluid equilibrium in the presence of a non-uniform density field.

Students often fall into the trap of blindly applying $P = \rho g h$ or assuming the force from the walls is zero.

A deeper trap is calculating the total weight of the fluid and incorrectly assuming it equals the force on the base, ignoring the downward normal force exerted by the sloped walls.

By forcing the integration of both the pressure gradient $dP/dy = \rho(y)g$ and the variable-density volume element $dV = A(y)dy$ , the problem discriminates between rote memorization of Pascal's principles and a fundamental understanding of free-body diagrams in continuous media.

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