GATE AE 2017 • Q44 NAT • 2 Marks

Flight Mechanics & Space Dynamics

Static Stability & Control

Subtopic: Vertical Tail Contribution to Directional Stability

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).

Official Answer Key:

0.18

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**: $C_{n\beta, v} = \eta_v \left(\frac{S_v l_v}{S b}\right) C_{L\alpha, v}$.

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