For constant free stream velocity and density, a change in lift for a large aspect ratio straight wing, with thin cambered airfoil section at small angles of attack, leads to
Options:
(A) a shift of the aerodynamic center and no shift of the center of pressure
(B) a shift of the center of pressure and no shift of the aerodynamic center
(C) shift of both the aerodynamic center and the center of pressure
(D) no shift either of the aerodynamic center or of the center of pressure
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2015 Q.12)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Linear Algebra (Eigenvalues & Eigenvectors)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> For constant free stream velocity and density, a change in lift for a large aspect ratio straight wing, with thin cambered airfoil section at small angles of attack, leads to
**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)** *"a shift of the aerodynamic center and no shift of the center of pressure"*: ✅ **CORRECT**.
- **(B)** *"a shift of the center of pressure and no shift of the aerodynamic center"*: ❌ **INCORRECT**.
- **(C)** *"shift of both the aerodynamic center and the center of pressure"*: ❌ **INCORRECT**.
- **(D)** *"no shift either of the aerodynamic center or of the center of pressure"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2015 (IIT Kanpur) master answer key with 100% precision.