Subtopic: Kutta-Joukowski Theorem & Thin Airfoil Theory
Question Statement:
For an airfoil section the pitching moment coefficient is determined about a
reference point that is 0.3 times the chord behind the leading edge. It varies with
the lift coefficient as shown in the table below. The distance of the aerodynamic
center from the leading edge of the airfoil as a fraction of the chord is ________
(rounded off to one decimal place).
cl 0.2 0.4 0.6 0.8
cm - 0.02 0 0.02 0.04
Official Answer Key:
0.2
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.54)**
**1. Intuitive Physical Setup & Given Parameters**:
Pitching moment coefficient $C_m$ is measured about a reference point located at $x_{\text{ref}} = 0.3 c$ behind the leading edge.
Data provided:
| $C_L$ | 0.2 | 0.4 | 0.6 | 0.8 |
| :---: | :---: | :---: | :---: | :---: |
| $C_m$ | -0.02 | 0.00 | +0.02 | +0.04 |
We need to find the location of the aerodynamic center $x_{\text{ac}} / c$ from the leading edge.
**2. Step-by-Step Aerodynamic Derivation**:
1. **Definition of Aerodynamic Center ($x_{\text{ac}}$)**:
By definition, the aerodynamic center is the point about which the pitching moment coefficient is independent of the lift coefficient:
\frac{\partial C_{m,\text{ac}}}{\partial C_L} = 0
2. **Moment Transfer Equation**:
The moment coefficient about reference point $x_{\text{ref}}$ is related to the moment about the aerodynamic center by:
C_{m,\text{ref}} = C_{m,\text{ac}} + \left(\frac{x_{\text{ref}}}{c} - \frac{x_{\text{ac}}}{c}\right) C_L
3. **Take the Derivative with Respect to $C_L$**:
\frac{d C_{m,\text{ref}}}{d C_L} = \frac{x_{\text{ref}}}{c} - \frac{x_{\text{ac}}}{c}
4. **Compute the Slope from the Data**:
\frac{d C_{m,\text{ref}}}{d C_L} = \frac{0.04 - (-0.02)}{0.8 - 0.2} = \frac{0.06}{0.6} = 0.10
5. **Solve for Aerodynamic Center**:
0.10 = 0.30 - \frac{x_{\text{ac}}}{c}
\frac{x_{\text{ac}}}{c} = 0.30 - 0.10 = 0.20
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $x_{\text{ac}} / c = 0.2$.
- **Rounding Requirement**: One decimal place $\implies \mathbf{0.2}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **0.2** (Acceptable Range: **0.2 to 0.2**)
- **Exam Takeaway**: $\frac{x_{\text{ac}}}{c} = \frac{x_{\text{ref}}}{c} - \frac{dC_{m,\text{ref}}}{dC_L}$. Because the slope is positive ($+0.1$), the aerodynamic center is ahead of the reference point: $0.3 - 0.1 = 0.2$.