For a conventional airplane with wing and horizontal tail, the stick-fixed static margin ($K_n$) is defined as $K_n = \frac{x_{np} - x_{cg}}{\bar{c}}$. If the aircraft is designed to be longitudinally statically stable in stick-fixed flight, which of the following conditions must strictly hold?

(A) $x_{cg} < x_{np}$ (Center of gravity lies ahead of the neutral point, so $K_n > 0$ and $\frac{dC_m}{dC_L} < 0$)

(B) $x_{cg} > x_{np}$ (Center of gravity lies behind the neutral point, so $K_n < 0$ and $\frac{dC_m}{dC_L} > 0$)

(C) $x_{cg} = x_{np}$ (Center of gravity coincides exactly with neutral point, neutral stability)

(D) $\frac{dC_m}{d\alpha} > 0$ with trim lift coefficient $C_{L,trim} = 0$

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Restoring Moment Condition:** A disturbance increasing $\alpha$ must create a nose-down pitch to restore trim ($dC_m/d\alpha < 0$). 2. **Stage 2: Pitching Moment Slope Relation:** $C_m = C_{m0} + \frac{dC_m}{dC_L} C_L$. Differentiating gives $\frac{dC_m}{dC_L} = \frac{x_{cg} - x_{np}}{\bar{c}} = -K_n$. 3. **Stage 3: Stability Inequality:** For stability, $\frac{dC_m}{dC_L} < 0 \implies -K_n < 0 \implies K_n > 0 \implies x_{cg} < x_{np}$. 4. **Stage 4: Distractor Analysis:** Option B leads to static instability (pitch-up divergence, requiring fly-by-wire). Option C is neutral stability ($K_n = 0$). 5. **Stage 5: Official Reference:** Bernard Etkin and Lloyd Duff Reid, *Dynamics of Flight: Stability and Control*, Chapter 3.

← Prev Question Next Question →