A model airplane has a horizontal tail volume ratio $V_H = 0.50$, horizontal tail efficiency $\eta_t = 0.90$, and elevator effectiveness parameter $\tau_e = \frac{\partial \alpha_t}{\partial \delta_e} = 0.40$. If the tail lift-curve slope is $a_t = 4.0\text{ rad}^{-1}$, the elevator control power derivative $C_{m_{\delta_e}} = \frac{\partial C_m}{\partial \delta_e}$ (in $\text{rad}^{-1}$) is closest to:

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### 1. Strategy & Theoretical Principles Elevator control power derivative $C_{m_{\delta_e}}$ quantifies the pitching moment change per unit deflection of the elevator $\delta_e$: $$C_{m_{\delta_e}} = -V_H \eta_t a_t \tau_e$$ where: - $V_H = \frac{S_t l_t}{S c}$ is the horizontal tail volume ratio - $\eta_t$ is the tail dynamic pressure ratio - $a_t$ is the tail lift-curve slope - $\tau_e = \frac{\partial \alpha_t}{\partial \delta_e}$ is the elevator angle-of-attack effectiveness factor. ### 2. Analytical Formulation & Calculation Given: - $V_H = 0.50$ - $\eta_t = 0.90$ - $a_t = 4.0\text{ rad}^{-1}$ - $\tau_e = 0.40$ $$C_{m_{\delta_e}} = -(0.50) \times (0.90) \times (4.0) \times (0.40)$$ $$C_{m_{\delta_e}} = -0.45 \times 1.60 = -0.720\text{ rad}^{-1}$$ ### 3. Distractor Forensics - Positive downward deflection of elevator ($\\delta_e > 0$) generates upward lift on the horizontal tail, which exerts a nose-down (negative) pitching moment. Hence $C_{m_{\delta_e}}$ is strictly negative. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Always keep the negative sign: $C_{m_{\delta_e}} < 0$. - **Standard Reference:** Nelson, R.C., *Flight Stability and Automatic Control*, Chapter 2.

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