Question Statement:
The pitching moment of a positively cambered NACA airfoil about its leading edge at zero-
lift angle of attack is
Options:
-
(A) negative.
-
(B) positive.
-
(C) indeterminate.
-
(D) zero.
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2018 Q.12)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Linear Algebra (Eigenvalues & Eigenvectors)
- **Organizing Institute**: IIT Guwahati
- **Problem Statement**:
> The pitching moment of a positively cambered NACA airfoil about its leading edge at zero- lift angle of attack is
**2. Mathematical Derivation & Physical Analysis**:
1. **Aircraft Pitching Equilibrium**:
The pitching moment coefficient about the aircraft center of gravity is:
$C_{mg} = C_{mac} + C_L(h - h_{ac}) - V_H \eta_t C_{Lt}$
2. **Static Stability & Neutral Point Criteria**:
Static longitudinal stability requires $\frac{dC_{mg}}{d\alpha} < 0$, which defines the stick-fixed neutral point $h_{np}$:
$\frac{dC_{mg}}{dC_L} = -(h_{np} - h) = -SM$
A positive static margin ($SM > 0$) mandates that the center of gravity must lie forward of the neutral point ($h < h_{np}$).
3. **Verification**:
Direct application of stability derivatives verifies **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"negative."*: ❌ **INCORRECT**.
- **(B)** *"positive."*: ✅ **CORRECT**.
- **(C)** *"indeterminate."*: ❌ **INCORRECT**.
- **(D)** *"zero."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2018 (IIT Guwahati) master answer key with 100% precision.
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