Subtopic: Longitudinal Trim and Static Stability Criteria
Question Statement:
For a conventional statically stable airplane, which one of the following combinations of pitching moment parameters about the center of gravity is CORRECT?
Options:
(A) $C_{m0} < 0$ and $C_{m\alpha} < 0$
(B) $C_{m0} > 0$ and $C_{m\alpha} < 0$
(C) $C_{m0} > 0$ and $C_{m\alpha} > 0$
(D) $C_{m0} < 0$ and $C_{m\alpha} > 0$
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.23 / AE Q.13)**
**1. Intuitive Physical Setup & Given Parameters**:
- Longitudinal static stability and trim equilibrium for an aircraft in level flight.
- Pitching moment equation: $C_m = C_{m0} + C_{m\alpha} \alpha$.
**2. Step-by-Step Aerodynamic Derivation**:
1. **Condition for Static Stability**:
Any pitch disturbance $\Delta \alpha > 0$ must generate a restoring pitch-down moment $\Delta C_m < 0$:
$C_{m\alpha} = \frac{\partial C_m}{\partial \alpha} < 0$
2. **Condition for Trim at Positive Angle of Attack ($\alpha_{trim} > 0$)**:
At trim equilibrium:
$C_m = 0 \implies C_{m0} + C_{m\alpha} \alpha_{trim} = 0$
$\alpha_{trim} = -\frac{C_{m0}}{C_{m\alpha}}$
Since for normal flight $\alpha_{trim} > 0$ and $C_{m\alpha} < 0$:
$-\frac{C_{m0}}{C_{m\alpha}} > 0 \implies C_{m0} > 0$
3. Both conditions require: $C_{m0} > 0$ and $C_{m\alpha} < 0$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** ($C_{m0} < 0, C_{m\alpha} < 0$): ❌ Cannot trim at positive $\alpha$.
- **(B)** ($C_{m0} > 0$ and $C_{m\alpha} < 0$): ✅ **CORRECT**. Classic longitudinal static stability and trim condition.
- **(C)** ($C_{m0} > 0, C_{m\alpha} > 0$): ❌ Statically unstable in pitch.
- **(D)** ($C_{m0} < 0, C_{m\alpha} > 0$): ❌ Statically unstable.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: Longitudinal static stability requires $C_{m0} > 0$ and $C_{m\alpha} < 0$.
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