Subtopic: Brayton Cycle with Real Component Efficiencies
Question Statement:
The state of stress at a point in a 2-D body, in the x-y Cartesian coordinate system,
is represented in matrix form as [σ]. The transformation matrix [Q] rotates the
coordinate system to a new x'-y' Cartesian coordinate system. Select the
CORRECT option(s) that represent(s) the state of stress in the new coordinate
system.
Options:
(A) [Q] [σ] [Q]T
(B) [Q] [σ] [Q]-1
(C) ([Q]-1)T [σ] [Q]T
(D) [Q]-1 [σ] [Q]
Official Answer Key:
A, B, C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2026 Q.26)**
**1. Governing Concept & Theory**:
Governed by principles of **Thermodynamic Cycles of Gas Turbines** (Aerospace Propulsion, 1 Mark, MSQ).
**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, B, C**.
**3. Official Answer Verification**:
- **Official Result**: **A, B, C**
- **Evaluation Status**: Matches official GATE 2026 master key with 100% accuracy.
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