For a general aviation airplane, which one of the following has a destabilizing
effect on its static roll stability?
Options:
(A) Wing with a positive dihedral angle
(B) Fuselage with a high wing
(C) Fuselage with a low wing
(D) Swept back wing
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.23)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to identify which aircraft configuration feature has a **destabilizing effect** on lateral (static roll) stability ($C_{l\beta}$).
Options given:
- Wing with a positive dihedral angle
- Fuselage with a high wing
- Fuselage with a low wing
- Swept-back wing
**2. Step-by-Step Aircraft Stability Derivation**:
1. **Definition of Static Roll Stability**:
- Lateral stability requires a restoring rolling moment when the aircraft experiences sideslip $\beta$:
C_{l\beta} = \frac{\partial C_l}{\partial \beta} < 0
- A negative $C_{l\beta}$ is stabilizing (e.g., sideslip to the right $\beta > 0$ generates a left-rolling moment $C_l < 0$ to level the wings).
2. **Analysis of Aircraft Components on $C_{l\beta}$**:
- **Positive Dihedral ($\Gamma > 0$)**: In sideslip, the windward wing sees a higher effective angle of attack than the leeward wing, producing stabilizing roll ($\Delta C_{l\beta} < 0$). **Stabilizing**.
- **High-Wing Configuration**: Cross-flow around the fuselage in sideslip induces an upward flow angle over the windward wing root and a downward flow over the leeward wing root, increasing lift on the windward wing. Contributes $\Delta C_{l\beta} \approx -0.05$. **Stabilizing**.
- **Low-Wing Configuration**: Cross-flow around the fuselage in sideslip induces a downward flow over the windward wing and upward flow over the leeward wing, reducing lift on the windward wing and rolling the aircraft further into the sideslip. Contributes $\Delta C_{l\beta} > 0$. **Destabilizing**.
- **Wing Sweepback ($\Lambda > 0$)**: The windward wing sees a higher effective normal velocity component $V\cos(\Lambda - \beta)$ while the leeward wing sees $V\cos(\Lambda + \beta)$, producing stabilizing roll ($\Delta C_{l\beta} < 0$). **Stabilizing**.
3. Therefore, a **fuselage with a low wing** has a destabilizing effect.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) Wing with a positive dihedral angle**: ❌ **INCORRECT**. Strongly stabilizing contribution to $C_{l\beta}$.
- **(B) Fuselage with a high wing**: ❌ **INCORRECT**. Stabilizing due to fuselage aerodynamic interference (pendulum/high-wing effect).
- **(C) Fuselage with a low wing**: ✅ **CORRECT**. Destabilizing effect on roll stability, which is why low-wing airplanes require greater geometric dihedral to compensate.
- **(D) Swept back wing**: ❌ **INCORRECT**. Strongly stabilizing contribution to $C_{l\beta}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: High wing = stabilizing (equivalent to $+1^\circ$ to $+3^\circ$ dihedral); Low wing = destabilizing (requires extra positive geometric dihedral to offset).