Subtopic: Tsiolkovsky Rocket Equation & Specific Impulse
Question Statement:
For studying wing vibrations, a wing of mass M and finite dimensions has been
idealized by assuming it to be supported using a linear spring of equivalent
stiffness k and a torsional spring of equivalent stiffness
k as shown in the figure.
The centre of gravity (CG) of the wing idealized as an airfoil is marked
in the figure. The number of degree(s) of freedom for this idealized wing vibration
model is _________. (Answer in integer)
Official Answer Key:
2.0
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2023 Q.30)**
**1. Governing Concept & Theory**:
Governed by principles of **Rocket Propulsion** (Aerospace Propulsion, 1 Mark, NAT).
**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 **2.0**.
**3. Official Answer Verification**:
- **Official Result**: **2.0** (Acceptable Range: `[2.0, 2.0]`)
- **Evaluation Status**: Matches official GATE 2023 master key with 100% accuracy.
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