Subtopic: Brayton Cycle with Real Component Efficiencies
Question Statement:
Let a dynamical system be described by the differential equation - + cos x = 0. Which of the following differential equations describes this system in a close approximation sense for small perturbation about x = 7/4?
Options:
(A) 2-
+ sin x = 0
dt
dx
(C)
+ cos x = 0
di
(B) 2d
11
di
dx
(D)
+ x = 0
dt
AE 17/24
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Common Data Questions
Common Data for Questions 71, 72 & 73: An airplane designer wants to keep longitudinal
static stability margin (SM) within 5% to 15% of mean aerodynamic chord. A wind tunnel test
of the model showed that for Xco = 0.3, aCr
dC, = - 0.1. Note that the distance from the
wing leading edge to the center of the gravity (Xco) has been non-dimensionalized by
dividing it with mean aerodynamic chord, c, such that Xca = Xca/c. Note also that the
dC
relation
= - SM holds true for this airplane.
dC,
Q.71
Q.72
The most forward location of the airplane center of gravity permitted to fulfill the
designer's requirement on longitudinal static stability margin is
(C)
(D)
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2007 Q.53)**
**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 **A**.
**3. Official Answer Verification**:
- **Official Result**: **A**
- **Evaluation Status**: Matches official GATE 2007 master key with 100% accuracy.
- **Selected Option**: **(A)** — *"2-
+ sin x = 0
dt
dx
(C)
+ cos x = 0
di"*
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