Subtopic: Bragg's Law & X-ray Diffraction (Laue and Powder)
Question Statement:
A hoop of mass M and radius R rolls without slipping along a straight line
on a horizontal surface as shown in the figure. A point mass m slides without
friction along the inner surface of the hoop, performing small oscillations
about the mean position. The number of degrees of freedom of the system
(in integer) is
M
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Official Answer Key:
2 to 2
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.30 / PH Q.20)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Crystal Structure & Lattice Dynamics* (`Bragg's Law & X-ray Diffraction (Laue and Powder)`) | **Section**: *Section 7: Solid State Physics*
- **Question Type**: **NAT** | **Marks**: **1 Mark** | **Negative Marks**: **0**
- **Governing Principles**: Solved directly by applying fundamental laws of **Crystal Structure & Lattice Dynamics**.
**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 **2 to 2**.
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