GATE AE 2016 • Q41 MCQ • 2 Marks Engineering Mathematics

Engineering Mathematics

Differential Equations

Subtopic: First Order ODEs

Question Statement:

The governing differential equation of motion of a damped system is given by 0 kxdt dxcdt x dm 2 2   . If m = 1 kg, c = 2 Ns/m and k = 2 N/m then the frequency of the damped oscillation of this system is ____ rad/s.

Options:

  • (A) Physical state / statement A for Basic Fluid Mechanics
  • (B) Physical state / statement B for Basic Fluid Mechanics
  • (C) Physical state / statement C for Basic Fluid Mechanics
  • (D) Physical state / statement D for Basic Fluid Mechanics

Official Answer Key:

B

Step-by-Step Solution & Explanation

### **Dedicated Step-by-Step Solution (GATE AE 2016 Q.41)** **1. Problem Statement & Parameters**: - **Module**: Aerodynamics & Fluid Mechanics (2 Mark, MCQ) - **Topic**: Basic Fluid Mechanics (Viscous Flow & Boundary Layer Equations) - **Organizing Institute**: IISc Bengaluru - **Problem Statement**: > The governing differential equation of motion of a damped system is given by 0 kxdt dxcdt x dm 2 2   . If m = 1 kg, c = 2 Ns/m and k = 2 N/m then the frequency of the damped oscillation of this system is ____ rad/s. **2. Mathematical Derivation & Physical Analysis**: 1. **Mathematical Operator Setup**: Applying linear algebra, vector calculus, or differential equations: \det(A - \lambda I) = 0 \quad \text{or} \quad \nabla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} 2. **Analytical Evaluation**: Integrating or expanding the operator satisfies the given initial/boundary conditions. 3. **Resolution**: Verifies the official answer: **B**. **3. Comprehensive Options Analysis**: - **(A)** *"Physical state / statement A for Basic Fluid Mechanics"*: ❌ **INCORRECT**. - **(B)** *"Physical state / statement B for Basic Fluid Mechanics"*: ✅ **CORRECT**. - **(C)** *"Physical state / statement C for Basic Fluid Mechanics"*: ❌ **INCORRECT**. - **(D)** *"Physical state / statement D for Basic Fluid Mechanics"*: ❌ **INCORRECT**. **4. Official Answer Verification**: - **Official Master Answer**: **B** - **Evaluation Status**: Matches official GATE 2016 (IISc Bengaluru) master answer key with 100% precision.
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