For an aircraft to possess stick-fixed longitudinal static stability, which of the following conditions must be satisfied regarding pitching moment derivative $C_{m,\alpha}$ and center of gravity position $x_{cg}$ relative to the stick-fixed neutral point $x_{np}$?

(A) C_{m,\alpha} < 0 \quad \text{and} \quad x_{cg} < x_{np}

(B) C_{m,\alpha} > 0 \quad \text{and} \quad x_{cg} < x_{np}

(C) C_{m,\alpha} < 0 \quad \text{and} \quad x_{cg} > x_{np}

(D) C_{m,\alpha} = 0 \quad \text{and} \quad x_{cg} = x_{np}

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Restoring Moment Criterion:** Any disturbance $\Delta\alpha > 0$ must induce $\Delta C_m < 0$ to restore trim, requiring $\frac{dC_m}{d\alpha} = C_{m,\alpha} < 0$. 2. **Stage 2: Neutral Point Formulation:** $C_{m,\alpha} = C_{L,\alpha} \left(\frac{x_{cg} - x_{np}}{\bar{c}}\right)$. Since $C_{L,\alpha} > 0$, $C_{m,\alpha} < 0$ mandates $x_{cg} < x_{np}$. 3. **Stage 3: Selection:** Option A correctly combines both conditions.

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