Question Statement:
As a candidate for a vertical tail, which one of the following airfoil sections is appropriate?
Options:
-
(A) NACA 0012
-
(B) NACA 2312
-
(C) NACA 23012
-
(D) Clarke Y profile
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2015 Q.14)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Differential Equations (First Order ODEs)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> As a candidate for a vertical tail, which one of the following airfoil sections is appropriate?
**2. Mathematical Derivation & Physical Analysis**:
1. **Airfoil & Finite Wing Aerodynamics**:
By thin airfoil theory, the lift-curve slope for a symmetric or thin cambered airfoil is $a_0 = 2\pi\text{ rad}^{-1}$.
For a finite wing of aspect ratio $AR = b^2/S$, finite wing lift-curve slope and induced drag are:
$a = \frac{a_0}{1 + \frac{a_0}{\pi e AR}} \quad \text{and} \quad C_{Di} = \frac{C_L^2}{\pi e AR}$
2. **Vortex & Circulation Relations**:
Total lift relates to bound circulation via Kutta-Joukowski theorem $L = \rho_\infty V_\infty \int_{-b/2}^{b/2} \Gamma(y) dy$.
3. **Result Verification**:
Direct evaluation gives the official result: **A**.
**3. Comprehensive Options Analysis**:
- **(A)** *"NACA 0012"*: ✅ **CORRECT**.
- **(B)** *"NACA 2312"*: ❌ **INCORRECT**.
- **(C)** *"NACA 23012"*: ❌ **INCORRECT**.
- **(D)** *"Clarke Y profile"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2015 (IIT Kanpur) master answer key with 100% precision.
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