Before You Begin

Question #43493

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid spherical planet of mass $M$ and radius $R$ is rotating about its axis. At time $t=0$ , its angular velocity is $\omega_0$ and it experiences a constant angular acceleration $\alpha$ . A small block of mass $m$ is initially at rest at the center of the planet. Starting at $t=0$ , an external agent moves the block through a frictionless, straight radial tunnel to the equator such that the block maintains a constant radial speed $v$ . The block reaches the equator at time $t=T$ . What is the total work done by the external agent on the block in the non-inertial reference frame of the planet?

Options

A.
$\frac{GMm}{2R} - m R^2 \left( \frac{\omega_0^2}{2} + \frac{2\omega_0 \alpha T}{3} + \frac{\alpha^2 T^2}{4} \right)$
B.
$\frac{3GMm}{5R} - m R^2 \left( \frac{\omega_0^2}{2} + \frac{2\omega_0 \alpha T}{3} + \frac{\alpha^2 T^2}{4} \right)$
C.
$\frac{GMm}{2R} - m R^2 \left( \frac{\omega_0^2}{2} + \omega_0 \alpha T + \frac{\alpha^2 T^2}{2} \right)$
D.
$\frac{3GMm}{5R} - m R^2 \left( \frac{\omega_0^2}{2} + \omega_0 \alpha T + \frac{\alpha^2 T^2}{2} \right)$

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

Pseudo Force Directionality|Geostationary Satellite Constraints|Gravitational Self-Energy

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate a non-uniformly rotating reference frame and correctly apply the gradient of gravitational potential.

The primary trap lies in the gravitational work term: students frequently confuse the potential difference between the center and surface of a solid sphere ( $\frac{GMm}{2R}$ ) with the gravitational self-energy or binding energy formula ( $\frac{3GMm}{5R}$ ).

Additionally, the non-uniform rotation introduces an Euler force and a time-dependent centrifugal force.

A common mistake is either incorrectly assigning a work value to the tangential Euler force during a radial displacement, or failing to integrate the time-dependent centrifugal force over the path, instead erroneously using the final angular velocity to calculate the pseudo-potential energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Pseudo Force Directionality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free