Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
The rate of change of moment coefficient with respect to the angle of attack,
d
dC m , at half chord
point of a thin airfoil, as per approximations from the thin airfoil theory is
Options:
(A) π/4 radian-1
(B) π/2 radian-1
(C) π radian-1
(D) 2π radian-1
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2016 Q.31)**
**1. Problem Statement & Parameters**:
- **Module**: Aerospace Structures (1 Mark, MCQ)
- **Topic**: Strength of Materials & Elasticity (Stress and Strain Tensors & Generalized Hooke's Law)
- **Organizing Institute**: IISc Bengaluru
- **Problem Statement**:
> The rate of change of moment coefficient with respect to the angle of attack, d dC m , at half chord point of a thin airfoil, as per approximations from the thin airfoil theory is
**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: **C**.
**3. Comprehensive Options Analysis**:
- **(A)** *"π/4 radian-1"*: ❌ **INCORRECT**.
- **(B)** *"π/2 radian-1"*: ❌ **INCORRECT**.
- **(C)** *"π radian-1"*: ✅ **CORRECT**.
- **(D)** *"2π radian-1"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2016 (IISc Bengaluru) master answer key with 100% precision.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.