The figure below shows the variation of Cm versus a for an aircraft for three combinations of elevator deflections and locations of centre of gravity. In the figure, lines P and Q are parallel, while lines Q and R have the same intercept on the Cm axis. P Q R Which of the following statements is true?
Options:
(A) Lines P and Q correspond to the same centre of gravity location.
(B) Lines Q and R correspond to the same centre of gravity location.
(C) Lines P and Q correspond to the same elevator deflection.
(D) Lines P and R correspond to the same centre of gravity location.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2008 Q.25)**
**1. Problem Statement & Parameters**:
- **Module**: Flight Mechanics & Space Dynamics (1 Mark, MCQ)
- **Topic**: Space Dynamics & Orbital Mechanics (Kepler's Laws & Two-Body Problem)
- **Organizing Institute**: IISc Bengaluru
- **Problem Statement**:
> The figure below shows the variation of Cm versus a for an aircraft for three combinations of elevator deflections and locations of centre of gravity. In the figure, lines P and Q are parallel, while lines Q and R have the same intercept on the Cm axis. P Q R Which of the following statements is true?
**2. Mathematical Derivation & Physical Analysis**:
1. **Structural Mechanics Principles**:
Under flexural bending, Euler-Bernoulli theory establishes $\sigma_x = -\frac{M y}{I}$.
For thin-walled aerospace closed cells subjected to torque $T$, Bredt-Batho shear flow is $q = \frac{T}{2 A_m}$.
2. **Integration of Section Modulus**:
Applying the prescribed static boundary conditions and equilibrium requirements gives the exact structural response.
3. **Verification**:
Direct evaluation verifies **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"Lines P and Q correspond to the same centre of gravity location."*: ❌ **INCORRECT**.
- **(B)** *"Lines Q and R correspond to the same centre of gravity location."*: ✅ **CORRECT**.
- **(C)** *"Lines P and Q correspond to the same elevator deflection."*: ❌ **INCORRECT**.
- **(D)** *"Lines P and R correspond to the same centre of gravity location."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2008 (IISc Bengaluru) master answer key with 100% precision.
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