Question Statement:
A conventional aircraft was analyzed to estimate the contribution of the vertical tail to the directional stability derivative $C_{n\beta}$. The vertical tail area is 2 m$^2$, moment arm is 5 m, wing area is 20 m$^2$, and wing span is 10 m. If the lift curve slope of the vertical tail is 4.0 rad$^{-1}$ and tail dynamic pressure ratio is 0.9, the contribution of the vertical tail $C_{n\beta, v}$ is _________ rad$^{-1}$ (round off to 2 decimal places).
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.44)**
**1. Intuitive Physical Setup & Given Parameters**:
- Vertical tail area: $S_v = 2\text{ m}^2$
- Vertical tail moment arm: $l_v = 5\text{ m}$
- Wing area: $S = 20\text{ m}^2$
- Wing span: $b = 10\text{ m}$
- Vertical tail lift curve slope: $C_{L\alpha, v} = 4.0\text{ rad}^{-1}$
- Tail dynamic pressure ratio: $\eta_v = 0.9$
**2. Step-by-Step Flight Dynamics Derivation**:
1. **Vertical Tail Volume Ratio ($V_v$)**:
V_v = \frac{S_v l_v}{S b} = \frac{2 \times 5}{20 \times 10} = \frac{10}{200} = 0.05
2. **Vertical Tail Contribution to Directional Stability ($C_{n\beta, v}$)**:
In the absence of sidewash ($\sigma = 0$):
C_{n\beta, v} = \eta_v V_v C_{L\alpha, v}
3. **Evaluating Numerical Value**:
C_{n\beta, v} = 0.9 \times 0.05 \times 4.0 = 0.9 \times 0.20 = 0.18\text{ rad}^{-1}
This matches the official range **[0.17, 0.19] rad$^{-1}$**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Official Range**: **0.17 to 0.19 rad$^{-1}$**.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **0.18**
- **Key Formula**: $C_{n\beta, v} = \eta_v \left(\frac{S_v l_v}{S b}\right) C_{L\alpha, v}$.