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Question #42284

Difficulty: 8/10 | 🟢 Free

The Question

A satellite of initial rest mass $m_0$ is in a stable circular orbit of radius $r_0$ around a planet of mass $M$ ( $M \gg m_0$ ). The satellite carries a nuclear power source. Over a period of time, the reactor converts a mass defect $\Delta m$ into energy, which is radiated away isotropically in the satellite's rest frame. Assuming no other forces act on the system and ignoring relativistic velocity effects ( $v \ll c$ ), what is the final orbital radius $r_f$ and the final angular momentum $L_f$ of the satellite about the center of the planet?

Options

A.
$r_f = r_0,\; L_f = (m_0 - \Delta m)\sqrt{GM r_0}$
B.
$r_f = r_0 \left( \frac{m_0}{m_0 - \Delta m} \right)^2,\; L_f = m_0\sqrt{GM r_0}$
C.
$r_f = r_0 \left( \frac{m_0}{m_0 - \Delta m} \right),\; L_f = m_0\sqrt{GM r_0}$
D.
$r_f = r_0,\; L_f = m_0\sqrt{GM r_0}$

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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

Weightlessness in Orbit|Mass Defect vs. Binding Energy|Integration

Why This Question Is Tricky

Common Mistakes

This question tests a profound misconception regarding the conservation of angular momentum in variable-mass systems.

Students are heavily conditioned to assume that in central force fields, the angular momentum of the orbiting body is strictly conserved.

Consequently, they will equate initial and final angular momentum ( $L_i = L_f$ ) and incorrectly calculate a new orbital radius $r_f = r_0 [m_0 / (m_0 - \Delta m)]^2$ .

The trap lies in failing to recognize that isotropic mass loss in the satellite's rest frame means the radiated photons carry away both linear and angular momentum in the planet's frame.

Thus, the satellite's velocity and orbital radius remain completely unaffected, while its individual angular momentum decreases proportionally to its mass defect.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Weightlessness in Orbit
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free