For a longitudinally statically stable aircraft, which one of the following
represents the relationship between the coefficient of pitching moment about the
center of gravity
mcgC and absolute angle of attack
a ?
(Note: nose-up moment is positive.)
Options:
(A)
(B)
(C)
(D)
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2023 Q.19)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to identify the defining conditions on the pitching moment curve ($C_m$ vs $\alpha$) for a conventional aircraft to possess longitudinal static stability and trim.
**2. Step-by-Step Mathematical Derivation**:
1. **Static Longitudinal Stability Condition**:
Static stability requires that any perturbation in angle of attack produces a restoring pitching moment:
- An increase in $\alpha$ ($+\Delta\alpha$) must generate a nose-down restoring moment ($-\Delta C_m$).
- Therefore, the slope of the pitching moment curve must be **strictly negative**:
$C_{m\alpha} = \frac{dC_m}{d\alpha} < 0$
2. **Trim Capability Condition**:
At the trim angle of attack $\alpha_{\text{trim}} > 0$, the net pitching moment must be zero: $C_m(\alpha_{\text{trim}}) = 0$.
Since $C_m = C_{m0} + C_{m\alpha} \alpha$, setting $C_m = 0$ requires:
$\alpha_{\text{trim}} = -\frac{C_{m0}}{C_{m\alpha}}$
Since $\alpha_{\text{trim}} > 0$ and $C_{m\alpha} < 0$, the zero-lift pitching moment must be **positive**:
$C_{m0} > 0$
3. **Conclusion**:
A statically stable, trimmable aircraft requires both a positive zero-lift moment ($C_{m0} > 0$) and a negative slope ($C_{m\alpha} < 0$).
**3. Comprehensive Options & Distractor Analysis**:
- **(A) C_m0 < 0 and dC_m/dα > 0**: ❌ **INCORRECT**. Statically unstable (divergent pitch) and trims at negative angle of attack.
- **(B) C_m0 > 0 and dC_m/dα > 0**: ❌ **INCORRECT**. Statically unstable.
- **(C) C_m0 < 0 and dC_m/dα < 0**: ❌ **INCORRECT**. Statically stable, but trims at negative $\alpha$, which cannot sustain positive lift.
- **(D) C_m0 > 0 and dC_m/dα < 0**: ✅ **CORRECT**. Guarantees stable restoring moment ($C_{m\alpha} < 0$) and positive trim lift ($C_{m0} > 0$).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: The Golden Rule of Longitudinal Static Stability: $C_{m0} > 0$ (positive intercept for positive trim lift) and $C_{m\alpha} < 0$ (negative slope for pitch restoring tendency).
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