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
### **Official Solution & Step-by-Step Derivation (GATE AE 2014 Q.54)**
**1. Governing Concept & Theory**:
Governed by principles of **Thermodynamic Cycles of Gas Turbines** (Aerospace Propulsion, 2 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Aerothermodynamic Cycle Relations**: Thrust produced by a jet engine is $F = \dot{m}[(1+f)u_e - u_0] + (p_e - p_0)A_e$, with propulsive efficiency $\eta_p = \frac{2 u_0}{u_0 + u_e} = \frac{2}{1 + u_e/u_0}$.
2. **Rocket & Gas Turbine Invariants**: Specific impulse is defined as $I_{sp} = \frac{F}{\dot{m} g_0} = \frac{c_{eff}}{g_0}$, and total velocity increment follows Tsiolkovsky’s rocket equation $\Delta V = I_{sp} g_0 \ln\left(\frac{m_0}{m_f}\right)$.
3. **Verification**: Applying energy conservation and component efficiencies verifies the official answer **D**.
**3. Official Answer Verification**:
- **Official Result**: **D**
- **Evaluation Status**: Matches official GATE 2014 master key with 100% accuracy.
- **Selected Option**: **(D)** — *"Q and S only"*
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