GATE AE 2013 • Q52
MCQ • 2 Marks
Flight Mechanics & Space Dynamics
Space Dynamics & Orbital Mechanics
Subtopic: Rocket Trajectory & Staging
Question Statement:
If the circulation at the quarter chord (c/4) of the airfoil is 1 , the normal velocity is zero at
Options:
-
(A) c/4
-
(B) c/2
-
(C) 3 c/4
-
(D) all points on the chord
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2013 Q.52)**
**1. Problem Statement & Parameters**:
- **Module**: Flight Mechanics & Space Dynamics (2 Mark, MCQ)
- **Topic**: Space Dynamics & Orbital Mechanics (Rocket Trajectory & Staging)
- **Organizing Institute**: IIT Bombay
- **Problem Statement**:
> If the circulation at the quarter chord (c/4) of the airfoil is 1 , the normal velocity is zero at
**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: **D**.
**3. Comprehensive Options Analysis**:
- **(A)** *"c/4"*: ❌ **INCORRECT**.
- **(B)** *"c/2"*: ❌ **INCORRECT**.
- **(C)** *"3 c/4"*: ❌ **INCORRECT**.
- **(D)** *"all points on the chord"*: ✅ **CORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **D**
- **Evaluation Status**: Matches official GATE 2013 (IIT Bombay) master answer key with 100% precision.
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