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Question #42320

Difficulty: 8/10 | 🟢 Free

The Question

A non-uniform spherical planet of mass $M$ and radius $R$ has a radially dependent density profile $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ , where $r$ is the distance from the center. A small payload of mass $m$ ( $m \ll M$ ) is initially at rest at the exact center of the planet. The payload is then launched to infinity through a narrow radial shaft. Simultaneously, the planet undergoes a structural collapse under its own gravity, settling into a uniform sphere of radius $R/2$ . What is the total change in the gravitational potential energy of the system (planet + payload)?

Options

A.
$\frac{GM}{R} \left( 2m - \frac{16}{35} M \right)$
B.
$\frac{GM}{R} \left( \frac{3}{2}m - \frac{3}{5} M \right)$
C.
$\frac{GM}{R} \left( m - \frac{16}{35} M \right)$
D.
$\frac{GM}{R} \left( 2m + \frac{16}{35} M \right)$

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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

Potential Energy Reference|Gravitational Self-Energy|Center of Mass (COM) Frame

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous definition of gravitational potential and self-energy for non-uniform mass distributions.

A common trap is assuming the potential at the center of a non-uniform body follows the uniform sphere formula ( $-1.5 GM/R$ ) or simply using the surface potential ( $-GM/R$ ).

Furthermore, students often forget that the self-energy of the system is a dynamic quantity that changes during structural collapse.

By requiring the calculation of both the non-uniform self-energy integral and the internal potential field, the problem forces students to abandon point-mass approximations and evaluate the energy state relative to the object's own constituent parts.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Potential Energy Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free