The closed thin-walled rectangular channel shown in figure (i) is opened by
introducing a sharp cut at the center of the bottom edge, as shown in figure (ii).
Which one of the following statements is correct?
Options:
(A) Centroids of (i) and (ii) coincide while shear centers do not
(B) Shear centers of (i) and (ii) coincide while centroids do not
(C) Both centroids and shear centers of (i) and (ii) coincide
(D) Neither centroids nor shear centers of (i) and (ii) coincide
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2022 Q.22)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Atmospheric Flight Mechanics** (Climb, Glide, Ceiling & Turn Performance) within the *Flight Mechanics & Space Dynamics* curriculum (1 Mark, MCQ).
**Problem Statement**:
> The closed thin-walled rectangular channel shown in figure (i) is opened by introducing a sharp cut at the center of the bottom edge, as shown in figure (ii). Which one of the following statements is correct?
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Flight Mechanics & Space Dynamics. Formulating the mathematical balance relations:
$\text{Analytical State Balance} = 0$
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`A`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"Centroids of (i) and (ii) coincide while shear centers do not"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
- **(B)** *"Shear centers of (i) and (ii) coincide while centroids do not"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *"Both centroids and shear centers of (i) and (ii) coincide"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"Neither centroids nor shear centers of (i) and (ii) coincide"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2022 (IIT Kharagpur) key.
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