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Question #42274

Difficulty: 8/10 | 🟢 Free

The Question

A planet of initial mass $m_0$ is in a circular orbit of radius $r_0$ and period $T_0$ around a star of mass $M$ ( $M \gg m_0$ ). It moves through a uniform cosmic dust cloud, accreting dust at a constant rate $\frac{dm}{dt} = \alpha$ . The dust is initially at rest relative to the star. Additionally, the cloud exerts a viscous drag force on the planet given by $\vec{F}_d = -\beta \vec{v}$ , where $\vec{v}$ is the planet's velocity and $\beta$ is a positive constant. Assuming the orbit remains quasi-circular, what is the orbital period $T$ of the planet when its mass reaches $2m_0$ ?

Options

A.
$T_0 2^{-3(1 + \beta/\alpha)}$
B.
$T_0 2^{-3(\beta/\alpha)}$
C.
$T_0 2^{-3(1 + \beta/2\alpha)}$
D.
$T_0 2^{-\frac{3}{2}(1 + \beta/\alpha)}$

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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

Kepler’s Third Law (Time Period)|Viscosity and Stokes' Law|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of Newton's Second Law and angular momentum in variable mass systems.

A common trap for JEE Advanced students is incorrectly assuming $\vec{F}_{net} = m\vec{a}$ applies to variable mass, leading to an incorrect torque equation $\frac{d\vec{L}}{dt} = \frac{\alpha-\beta}{m}\vec{L}$ .

The correct fundamental law is $\vec{F}_{net} = \frac{d\vec{p}}{dt}$ .

Furthermore, students often forget that the accreted dust, being initially at rest, contributes zero momentum but increases the mass, effectively acting as an additional drag force (accretion drag).

Integrating the correct angular momentum decay with Kepler's Third Law for a shrinking quasi-circular orbit yields the true orbital period.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Kepler’s Third Law (Time Period)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free