Before You Begin

Question #42113

Difficulty: 8/10 | 🟢 Free

The Question

A massive, isolated spherical shell of an incompressible liquid has an inner radius $R$ and an outer radius $2R$ . The liquid has a uniform density $\rho$ and surface tension $S$ . The shell encloses a concentric spherical cavity containing a massless gas. The external region outside the shell is a vacuum ( $P_{out} = 0$ ). Assuming the system is in mechanical equilibrium under its own gravitational self-interaction and surface tension, what is the pressure $P_{in}$ of the gas inside the cavity? (Here $G$ is the universal gravitational constant)

Options

A.
$\frac{3S}{R} + \frac{4}{3}\pi G \rho^2 R^2$
B.
$\frac{3S}{R} + 2\pi G \rho^2 R^2$
C.
$\frac{4S}{R} + \frac{4}{3}\pi G \rho^2 R^2$
D.
$\frac{S}{R} + \frac{4}{3}\pi G \rho^2 R^2$

Check Your Answer

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

Excess Pressure|Gravitational Field Variation (Depth)|Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Laplace's Law, gravitational self-potential (via Gauss's Law), and hydrostatic equilibrium in a non-uniform field.

The primary trap lies in the miscalculation of the self-gravitational field by either ignoring the cavity (assuming $g \propto r$ as in a solid sphere) or failing to integrate the non-linear hydrostatic pressure.

Additionally, students frequently misapply the Young-Laplace equation, either by blindly using the standard soap bubble formula ( $\Delta P = 4S/R$ ) without accounting for the different radii of the inner and outer surfaces, or by assigning the wrong sign to the pressure jump at the concave inner boundary.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Excess Pressure (Laplace Law)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free