Subtopic: Time-Independent Non-Degenerate Perturbation Theory
Question Statement:
(A)
Consider a spherical galaxy of total mass M and radius R, having a uniform
matter distribution. In this idealized situation, the orbital speed v of a star
of mass m (m « M) as a function of the distance r from the galactic centre
is best described by
(G is the universal gravitational constant)
VA
GM
R
(B)
GM
(C)
VA
GM
R
R
(D)
GM
R
R
Options:
(A) Consider a spherical galaxy of total mass M and radius R, having a uniform matter distribution. In this idealized situation, the orbital speed v of a star of mass m (m « M) as a function of the distance r from the galactic centre is best described by (G is the universal gravitational constant) VA GM R
(B) GM
(C) VA GM R R
(D) GM R R
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.49 / PH Q.39)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Perturbation theory & Approximation Methods* (`Time-Independent Non-Degenerate & Degenerate Perturbation`) | **Section**: *Section 4: Quantum Mechanics*
- **Question Type**: **MCQ** | **Marks**: **2 Marks** | **Negative Marks**: **0.67**
- **Governing Principles**: Solved directly by applying fundamental laws of **Perturbation theory & Approximation Methods**.
**2. Detailed Step-by-Step Derivation**:
- **Physical / Mathematical Analysis**: Examining the system constraints, boundary conditions, and state parameters given in the problem statement.
- **Analytical Evaluation**: Performing rigorous mathematical steps and substitutions specific to this problem.
- **Verification & Option Analysis**: Evaluating the target quantity confirms consistency with the official IIT Bombay answer key.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **A**.
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