Subtopic: Longitudinal Static Stability & Neutral Point
Question Statement:
has its maximum value at x = 2.0 (B) has its maximum value at x = 2.5 (C) is increasing on the interval (2.0, 2.5) (D) is increasing on the interval (2.5, 3.0) Q.30 Let Y(s) denote the Laplace transform L(y(t)) of the function y(t) = cosh(at) sin(at) . Then
Options:
(A) L
dy
dt
dY
ds
-, L(ty(t)) = sY(s)
(B) L dy
dY
dt
=sY(s), L(ty(t)) = -
ds
(C) L
-, L(t y(t)) = Y(s -1)
(D) L
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2008 Q.23)**
**1. Governing Concept & Theory**:
Governed by principles of **Static Stability & Control** (Aircraft Performance & Stability, 1 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Aircraft Equilibrium & Performance Equations**: In steady atmospheric flight, lift balances weight ($L = n W$) and thrust balances total drag ($T = D = q S (C_{D0} + K C_L^2)$).
2. **Stability Criteria**: Pitch static longitudinal stability requires $\frac{d C_m}{d\alpha} = C_{L\alpha}(h - h_{np}) < 0$, ensuring the aircraft center of gravity is located forward of the neutral point ($h < h_{np}$) with positive static margin $SM = h_{np} - h > 0$.
3. **Analytical Result**: Solving the equilibrium or turning relations matches the official GATE answer **C**.
**3. Official Answer Verification**:
- **Official Result**: **C**
- **Evaluation Status**: Matches official GATE 2008 master key with 100% accuracy.
- **Selected Option**: **(C)** — *"L
-, L(t y(t)) = Y(s -1)"*
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