Subtopic: Dutch Roll, Spiral & Roll Subsidence Modes
Question Statement:
A cantilever beam of length 2a is loaded at the tip with force F as shown in the
figure. The beam is supported in the middle by a roller with a pin. The magnitude
of moment reaction at the built -in end of the beam is
Fa, where
is ______ .
(round off to one decimal place)
Official Answer Key:
0.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2023 Q.50)**
**1. Governing Concept & Theory**:
Governed by principles of **Dynamic Stability** (Aircraft Performance & Stability, 2 Mark, NAT).
**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 **0.5**.
**3. Official Answer Verification**:
- **Official Result**: **0.5** (Acceptable Range: `[0.5, 0.5]`)
- **Evaluation Status**: Matches official GATE 2023 master key with 100% accuracy.
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