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Question #42292

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly spherical planet of mass $M$ and radius $R$ rotates with a constant angular velocity $\omega$ . A particle of mass $m$ is initially at rest at the North Pole of the planet. An external agent slowly transports the particle along the planet's surface to the equator, and then lifts it radially to a synchronous orbit of radius $R_s$ (where the particle's orbital angular velocity matches the planet's rotation). In the rotating frame of the planet, what is the total work done by the external agent? (Assume no other celestial bodies exert gravitational influence).

Options

A.
$\frac{GMm}{R} - \frac{3}{2} m \omega^2 R_s^2$
B.
$\frac{GMm}{R} - \frac{3}{2} m \omega^2 R_s^2 + \frac{1}{2} m \omega^2 R^2$
C.
$\frac{GMm}{R} - \frac{1}{2} m \omega^2 R_s^2$
D.
$\frac{GMm}{R} - m \omega^2 R_s^2$

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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

Effect of Earth's Rotation on g|State Function vs Path Function|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of the effective potential field in a rotating non-inertial frame.

The primary trap is assuming that moving a mass along the surface of the planet from the pole to the equator requires zero work, which incorrectly treats the effective gravity as a constant scalar or ignores the latitude dependence of the centrifugal potential.

Students falling for this will calculate the work only for the radial lift, leading to an extra $+\frac{1}{2}m\omega^2 R^2$ term.

Another trap is calculating the change in total mechanical energy in the inertial frame without accounting for the work done by the planet's surface to keep the particle co-rotating, which yields the incorrect $-\frac{1}{2}m\omega^2 R_s^2$ term.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Effect of Earth's Rotation on g
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free