Consider a finite wing of aspect ratio 10 with span effectiveness factor 0.95. Its
airfoil section has a lift slope of 0.106 per degree and a zero-lift angle of attack of
-1.5°. The lift coefficient of the wing at an angle of attack of 3.5° is ________
(rounded off to 2 decimal places).
Official Answer Key:
0.44
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.56)**
**1. Intuitive Physical Setup & Given Parameters**:
- Finite wing aspect ratio: $AR = 10$
- Span effectiveness factor: $e = 0.95$
- Airfoil lift slope: $a_0 = 0.106\text{ per degree}$
- Zero-lift angle of attack: $\alpha_{L=0} = -1.5^\circ$
- Operating angle of attack: $\alpha = 3.5^\circ$
We need to calculate the wing lift coefficient $C_L$ (rounded off to 2 decimal places).
**2. Step-by-Step Aerodynamic Derivation**:
1. **Finite Wing Lift Curve Slope Formula**:
Using Helmbold/Prandtl finite wing theory, the 3D lift curve slope $a$ (per degree) is given by:
$a = \frac{a_0}{1 + \frac{57.3 a_0}{\pi AR e}}$
where the factor $57.3 \approx \frac{180}{\pi}$ converts radians to degrees in the induced downwash term.
2. **Compute Denominator Term**:
$\frac{57.3 \times 0.106}{\pi \times 10 \times 0.95} = \frac{6.0738}{29.8451} \approx 0.20351$
$a = \frac{0.106}{1 + 0.20351} = \frac{0.106}{1.20351} \approx 0.088076\text{ per degree}$
3. **Calculate Lift Coefficient $C_L$**:
The lift coefficient is:
$C_L = a (\alpha - \alpha_{L=0})$
$\alpha - \alpha_{L=0} = 3.5^\circ - (-1.5^\circ) = 5.0^\circ$
$C_L = 0.088076 \times 5.0 \approx 0.44038 \approx 0.44$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $C_L \approx 0.440$.
- **Rounding Requirement**: Two decimal places $\implies \mathbf{0.44}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **0.44** (Acceptable Range: **0.42 to 0.46**)
- **Exam Takeaway**: When airfoil slope is given per degree, convert the induced drag correction term via $\frac{180 a_0}{\pi^2 AR e} \approx \frac{57.3 a_0}{\pi AR e}$ before computing $C_L = a(\alpha - \alpha_0)$.
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