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Question #42488

Difficulty: 8/10 | 🟢 Free

The Question

A spherically symmetric planet of radius $R$ and total mass $M$ has a non-uniform density distribution $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ for $r \le R$ . A satellite of mass $m$ is moving in a circular orbit of radius $r = \frac{R}{2}$ inside a narrow equatorial tunnel. What is the minimum additional energy required for the satellite to escape the planet's gravitational field to infinity?

Options

A.
$\frac{21GMm}{16R}$
B.
$\frac{13GMm}{8R}$
C.
$\frac{11GMm}{16R}$
D.
$\frac{27GMm}{16R}$

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

Mechanical Energy Conservation|Gravitational Potential Energy (System)|Orbital Velocity Energy Balance

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to derive gravitational potential and orbital energy for a non-uniform density distribution, a classic JEE Advanced trap.

Students commonly make two fatal errors: assuming the potential inside the planet follows the point-mass formula $V(r) = -GM/r$ , or incorrectly applying the uniform sphere potential formula.

By defining $\rho(r)$ , the problem forces the derivation of the enclosed mass $M(r)$ and the subsequent integration of the gravitational field from infinity to $r=R/2$ to find the true potential $V(r)$ .

Furthermore, students might forget to account for the satellite's existing orbital kinetic energy, leading them to calculate only the escape work from rest.

Quick Facts About This Question

Question Information

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