Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.48)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Atmospheric Flight Mechanics** (Range and Endurance of Jet & Propeller Aircraft) within the *Flight Mechanics & Space Dynamics* curriculum (2 Mark, MSQ).
**Problem Statement**:
> For an airplane having directional / weathercock static stability, which of the following options is/are correct?
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Flight Mechanics & Space Dynamics. Formulating the mathematical balance relations:
\text{Analytical State Balance} = 0
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`A, B, C`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"The airplane when disturbed in yaw , from an equilibrium state, will experience a
restoring moment"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(B)** *"The variation of yawing moment coefficient (Cn) with
sideslip angle ($\beta$) for the airplane will look like"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(C)** *"The airplane will always tend to point into the relative wind"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(D)** *"The airplane when disturbed in yaw will return to equilibrium state in a finite
amount of time after removing the disturbance"*: ❌ **INCORRECT**. Contradicts governing physical laws.
*Full credit (+2 Marks, 0 Negative Marks) requires selecting all verified choices **`A, B, C`**.*
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B, C**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2024 (IISc Bengaluru) key.