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Question #42095

Difficulty: 8/10 | 🟢 Free

The Question

A hollow rigid sphere of inner radius $R$ is completely filled with an incompressible liquid of density $\rho$ . The sphere rotates with a constant angular velocity $\omega$ about a vertical diameter. A small hole at the highest point of the sphere keeps the liquid pressure there equal to the atmospheric pressure $P_0$ . Assuming the acceleration due to gravity is $g$ , what is the net upward force exerted by the liquid on the upper half of the sphere?

Options

A.
$\pi R^2 P_0 + \frac{1}{3} \pi \rho g R^3 + \frac{\pi}{4} \rho \omega^2 R^4$
B.
$\pi R^2 P_0 + \frac{2}{3} \pi \rho g R^3 + \frac{\pi}{4} \rho \omega^2 R^4$
C.
$\pi R^2 P_0 + \frac{1}{3} \pi \rho g R^3 + \frac{\pi}{2} \rho \omega^2 R^4$
D.
$\pi R^2 P_0 + \frac{2}{3} \pi \rho g R^3 + \frac{\pi}{2} \rho \omega^2 R^4$

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

Centripetal Acceleration Vector|Accelerated Frame Hydrostatics|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of hydrostatics in an accelerated reference frame, specifically targeting the trap of 'Pressure Gradient Fluctuation'.

Students often incorrectly assume that the pressure at the equatorial plane is uniform and equal to its maximum value at the edge ( $P_0 + \rho g R + \frac{1}{2}\rho \omega^2 R^2$ ), leading them to multiply this by the total area $\pi R^2$ .

This ignores the radial pressure gradient $\partial P/\partial r = \rho \omega^2 r$ , which requires integration over the circular cross-section.

Furthermore, students frequently miscalculate the net vertical force from the static fluid by confusing the Archimedes buoyant force with the actual integrated pressure force, incorrectly using the hemisphere's weight $\frac{2}{3}\pi R^3 \rho g$ as the sole gravitational contribution instead of evaluating the boundary equilibrium $F_{up} = F_{eq} - W$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Centripetal Acceleration Vector
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free