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Question #42266

Difficulty: 8/10 | 🟢 Free

The Question

Two isolated stars of masses $m$ and $2m$ interact only via their mutual gravitational force. In the center of mass reference frame, the system has a constant total angular momentum $L$ . The relative radial motion of the stars can be modeled as a single particle of equivalent mass moving in a 1D effective potential $V_{eff}(r)$ , where $r$ is the distance between the stars. What is the radius $r_c$ of the stable circular orbit and the corresponding minimum effective potential $V_{min}$ ?

Options

A.
$r_c = \frac{3L^2}{4Gm^3}, \; V_{min} = -\frac{4G^2m^5}{3L^2}$
B.
$r_c = \frac{L^2}{2Gm^3}, \; V_{min} = -\frac{2G^2m^5}{L^2}$
C.
$r_c = \frac{L^2}{6Gm^3}, \; V_{min} = -\frac{6G^2m^5}{L^2}$
D.
$r_c = \frac{3L^2}{2Gm^3}, \; V_{min} = -\frac{2G^2m^5}{3L^2}$

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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

Gravitational Potential Reference|Kepler’s Second Law|Center of Mass (COM) Frame

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from a two-body system to an equivalent one-body problem in the center of mass frame.

A common trap is failing to correctly identify the reduced mass $\mu$ for the kinetic energy (centrifugal) term, often mistakenly using the total mass $M$ or the mass of a single body.

Furthermore, students frequently forget that the effective potential $V_{eff}(r)$ must incorporate the centrifugal potential energy $L^2/(2\mu r^2)$ alongside the true gravitational potential $-Gm_1m_2/r$ .

Minimizing this combined effective potential is required to find the stable circular orbit parameters.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Gravitational Potential Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free