Subtopic: Free & Forced Vibrations of Continuous Beams
Question Statement:
A uniform rigid bar of mass 3 kg is hinged at point F, and supported by a spring of
stiffness k=100 N/m, as shown in the figure. The natural frequency of free
vibration of the system is ____rad/s (answer in integer).
Official Answer Key:
10.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.57)**
**1. Intuitive Physical Setup & Given Parameters**:
- Uniform rigid bar of mass $m = 3\text{ kg}$ and length $L$, hinged at one end $F$.
- Supported at the other end by a spring of stiffness $k = 100\text{ N/m}$.
We need to find the natural frequency of free vibration $\omega_n$ in $\text{rad/s}$ (integer).
**2. Step-by-Step Structural Dynamics Derivation**:
1. **Mass Moment of Inertia About Hinge $F$**:
For a uniform rigid bar of mass $m$ hinged at one end:
I_F = \frac{1}{3} m L^2 = \frac{1}{3} (3) L^2 = L^2\text{ kg}\cdot\text{m}^2
2. **Equation of Motion for Small Angular Oscillations $\theta$**:
Taking moments about hinge $F$:
I_F \ddot{\theta} + \sum M_{\text{restoring}} = 0
The spring attached at the end of the bar undergoes deflection $x = L\theta$, exerting restoring force $F_s = k L \theta$ with moment arm $L$:
M_{\text{spring}} = k L^2 \theta
I_F \ddot{\theta} + k L^2 \theta = 0
\left(\frac{1}{3} m L^2\right) \ddot{\theta} + k L^2 \theta = 0
3. **Cancel $L^2$ and Solve for $\omega_n$**:
\ddot{\theta} + \frac{3k}{m} \theta = 0
\omega_n = \sqrt{\frac{3k}{m}}
4. **Substitute Numerical Values**:
- $k = 100\text{ N/m}$
- $m = 3\text{ kg} $\omega_n = \sqrt{\frac{3 \times 100}{3}} = \sqrt{100} = 10\text{ rad/s}
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $\omega_n = 10\text{ rad/s}$.
- **Rounding Requirement**: Nearest integer $\implies \mathbf{10}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **10.0** (Acceptable Range: **10 to 10**)
- **Exam Takeaway**: Hinged bar natural frequency is $\omega_n = \sqrt{\frac{k L^2}{I_F}} = \sqrt{\frac{k L^2}{\frac{1}{3}mL^2}} = \sqrt{\frac{3k}{m}} = \sqrt{300/3} = 10\text{ rad/s}$.