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Question #42258

Difficulty: 8/10 | 🟢 Free

The Question

A massive spherical cloud of dust of radius $R$ has a non-uniform mass density $\rho(r) = \rho_0 \frac{R}{r}$ for $r \le R$ , where $\rho_0$ is a positive constant and $r$ is the radial distance from the center. A particle of mass $m$ is initially moving in a circular orbit of radius $r_0$ ( $r_0 < R$ ) inside the cloud. The particle is given a small radial impulse, causing it to undergo small radial oscillations. What is the angle $\Delta \Theta$ swept by the particle's radius vector between two successive points of minimum radial distance?

Options

A.
$\frac{2\pi}{\sqrt{3}}$
B.
$\pi$
C.
$2\pi$
D.
$\frac{\pi}{\sqrt{3}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Shell Theorem (Internal vs External)|Angular Momentum conservation|Kepler’s Second Law (Areal Velocity)

Why This Question Is Tricky

Common Mistakes

This question tests the dynamics of orbits in non-Keplerian potentials.

A common trap is assuming that all central forces produce closed orbits where the angle between successive periapses is $2\pi$ (as in a $1/r^2$ force) or $\pi$ (as in a uniform density sphere where $F \propto r$ ).

By introducing a non-uniform density $\rho \propto 1/r$ , the gravitational field becomes constant.

Students must correctly apply the Shell Theorem to find the non-standard potential, construct the effective potential $U_{eff}$ , and use small oscillation theory (Bertrand's Theorem concepts) to find the ratio of radial to angular frequencies.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Shell Theorem (Internal vs External)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free