Consider the following four statements regarding aircraft longitudinal stability:
P. CM,cg at zero lift must be positive
Q. $\partial$CM,cg/$\partial \alpha$a must be negative ($\alpha$a is absolute angle of attack)
R. CM,cg at zero lift must be negative
S. Slope of CL versus $\alpha$a must be negative
Which of the following combination is the necessary criterion for stick fixed longitudinal balance
and static stability?
Options:
(A) Q and R only
(B) Q, R, and S only
(C) P and Q only
(D) Q and S only
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2014 Q.54)**
**1. Problem Statement & Parameters**:
- **Module**: Propulsion (2 Mark, MCQ)
- **Topic**: Thermodynamic Cycles of Gas Turbines (Thrust Equations & Specific Fuel Consumption)
- **Organizing Institute**: IIT Kharagpur
- **Problem Statement**:
> Consider the following four statements regarding aircraft longitudinal stability: P. CM,cg at zero lift must be positive Q. $\partial$CM,cg/$\partial \alpha$a must be negative ($\alpha$a is absolute angle of attack) R. CM,cg at zero lift must be negative S. Slope of CL versus $\alpha$a must be negative Which of the following combination is the necessary criterion for stick fixed longitudinal balance and static stability?
**2. Mathematical Derivation & Physical Analysis**:
1. **Aircraft Equilibrium & Stability Derivatives**:
The aircraft pitching moment coefficient about the center of gravity is:
C_{mg} = C_{mac} + C_L(h - h_{ac}) - V_H \eta_t C_{Lt}
2. **Neutral Point & Static Margin**:
The stick-fixed neutral point location is $h_{np} = h_{ac} + V_H \eta_t \frac{a_t}{a}(1 - \frac{d\epsilon}{d\alpha})$.
Static stability requires $h < h_{np}$, with static margin $SM = h_{np} - h > 0$.
3. **Verification**:
Applying trim and stability equations matches **D**.
**3. Comprehensive Options Analysis**:
- **(A)** *"Q and R only"*: ❌ **INCORRECT**.
- **(B)** *"Q, R, and S only"*: ❌ **INCORRECT**.
- **(C)** *"P and Q only"*: ❌ **INCORRECT**.
- **(D)** *"Q and S only"*: ✅ **CORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **D**
- **Evaluation Status**: Matches official GATE 2014 (IIT Kharagpur) master answer key with 100% precision.