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Question #43652

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The Question

A hypothetical 'gravitational atom' consists of a particle of mass $m$ orbiting a central star of mass $M$ ( $M \gg m$ ) under purely gravitational forces. The system's energy levels are quantized according to Bohr's postulates. Let $R_0$ be the magnitude of the initial ground state energy of this system. The central star slowly and isotropically loses mass due to radiation over a very long time, until its mass becomes $M/2$ . The process is slow enough that the orbital quantum number $n$ remains an adiabatic invariant. Assuming the emitted radiation does not interact with the orbiting particle, find the new ground state binding energy $E'_b$ and the work $W_g$ done by the gravitational force on the particle during this entire process.

Options

A.
$E'_b = \frac{R_0}{4}, \; W_g = -\frac{3}{4} R_0$
B.
$E'_b = \frac{R_0}{4}, \; W_g = \frac{3}{4} R_0$
C.
$E'_b = \frac{R_0}{2}, \; W_g = -\frac{1}{2} R_0$
D.
$E'_b = \frac{R_0}{4}, \; W_g = -\frac{3}{2} R_0$

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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

Rydberg Constant Sensitivity|Orbital Velocity Energy Balance|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the work-energy theorem and the concept of adiabatic invariants in a Bohr-quantized system.

The primary trap is the misapplication of potential energy in a time-varying field.

Students commonly assume that the work done by gravity is $W_g = -\Delta U$ or $W_g = \Delta E$ .

However, because the central mass $M(t)$ is decreasing, the gravitational potential field is explicitly time-dependent.

In such fields, the standard relation $W = -\Delta U$ fails because the potential energy changes even when the particle is stationary.

The rigorous approach relies on the fundamental work-energy theorem for a single particle, $W_{net} = \Delta K$ .

Furthermore, students must correctly derive that the energy levels of a gravitational Bohr atom scale with $M^2$ , not $M$ , to find the correct final state.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Rydberg Constant Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free