Subtopic: Longitudinal Static Stability & Neutral Point
Question Statement:
To achieve longitudinal static stability of a general aviation airplane, which one of the following conditions should be satisfied?
Options:
(A) The center of gravity of the airplane should be aft of the neutral point.
(B) The center of gravity of the airplane should be forward of the neutral point.
(C) The stability coefficient $\frac{\partial C_m}{\partial \alpha}$ ($C_m$ is the airplane pitching moment coefficient and $\alpha$ is the angle of attack) is positive.
(D) The static margin is negative.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.25)**
**1. Intuitive Physical Setup & Given Parameters**:
- **Physical Concept**: Longitudinal static pitch stability of an aircraft requires that any disturbance in pitch produces a restoring pitching moment to return the aircraft to its equilibrium trim state.
- **Governing Variable**: Static margin $K_n = \frac{x_{np} - x_{cg}}{\bar{c}}$, where $x_{np}$ is the longitudinal position of the aerodynamic neutral point and $x_{cg}$ is the center of gravity location.
**2. Step-by-Step Mathematical Derivation**:
1. **Pitching Moment Stability Derivative**:
For an aircraft disturbed by an incremental angle of attack $\Delta \alpha > 0$, static stability requires a pitch-down (restoring) moment:
C_{m\alpha} = \frac{\partial C_m}{\partial \alpha} < 0
2. **Neutral Point and Static Margin Formulation**:
The pitching moment about the center of gravity relates to the neutral point by:
C_{m\alpha} = -C_{L\alpha} \left( \frac{x_{np} - x_{cg}}{\bar{c}} \right)
Since the lift curve slope of the complete aircraft $C_{L\alpha} > 0$:
C_{m\alpha} < 0 \iff x_{np} - x_{cg} > 0 \implies x_{cg} < x_{np}
Therefore, the center of gravity $x_{cg}$ must lie forward (ahead) of the neutral point $x_{np}$, yielding a positive static margin:
\text{Static Margin} = \frac{x_{np} - x_{cg}}{\bar{c}} > 0
**3. Comprehensive Options & Distractor Analysis**:
- **Option (A)** (aft of neutral point): ❌ **Incorrect**. If $x_{cg} > x_{np}$, the static margin is negative and $C_{m\alpha} > 0$, making the aircraft statically unstable in pitch (pitch-up divergence).
- **Option (B)** (forward of neutral point): ✅ **CORRECT**. $x_{cg} < x_{np}$ ensures a positive static margin and a negative restoring pitching moment slope $C_{m\alpha} < 0$.
- **Option (C)** ($C_{m\alpha} > 0$): ❌ **Incorrect**. Positive $C_{m\alpha}$ means pitch divergence, not stability.
- **Option (D)** (negative static margin): ❌ **Incorrect**. Negative static margin indicates instability.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: For longitudinal static stability: $x_{cg} < x_{np}$ (cg forward of neutral point), static margin $> 0$, and $\frac{\partial C_m}{\partial \alpha} < 0$.