The static margin of an airplane with mean aerodynamic chord $c$ is defined as $K_n = \frac{x_{np} - x_{cg}}{c}$, where $x_{np}$ is the neutral point and $x_{cg}$ is the center of gravity location. If $K_n = +0.10$ and the aircraft lift-curve slope is $C_{L_\alpha} = 5.0\text{ rad}^{-1}$, the static pitch stability derivative $C_{m_\alpha} = \frac{\partial C_m}{\partial \alpha}$ (in $\text{rad}^{-1}$) is:

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### 1. Strategy & Theoretical Principles From classical longitudinal static stability theory, the pitching moment derivative about the center of gravity is related directly to the static margin $K_n$ by: $$C_{m_\alpha} = -C_{L_\alpha} K_n = -C_{L_\alpha} \left(\frac{x_{np} - x_{cg}}{c}\right)$$ For positive static longitudinal stability, the center of gravity must lie ahead of the neutral point ($x_{cg} < x_{np}$), making $K_n > 0$ and $C_{m_\alpha} < 0$. ### 2. Analytical Formulation & Calculation Given: - $K_n = +0.10$ - $C_{L_\alpha} = 5.0\text{ rad}^{-1}$ $$C_{m_\alpha} = -(5.0) \times (0.10) = -0.50\text{ rad}^{-1}$$ ### 3. Distractor Forensics - Candidate omitting the minus sign reports $+0.50\text{ rad}^{-1}$, which represents a statically unstable airplane. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Longitudinal stability requires a negative derivative: $C_{m_\alpha} < 0$. - **Standard Reference:** Nelson, R.C., *Flight Stability and Automatic Control*, Chapter 2.

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