Subtopic: Thrust Equations & Specific Fuel Consumption
Question Statement:
Consider the following four statements regarding aircraft longitudinal stability:
P. CM,cg at zero lift must be positive
Q. ∂CM,cg/∂αa must be negative (αa is absolute angle of attack)
R. CM,cg at zero lift must be negative
S. Slope of CL versus αa 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. ∂CM,cg/∂αa must be negative (αa is absolute angle of attack) R. CM,cg at zero lift must be negative S. Slope of CL versus αa 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.
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