A uniform rigid prismatic bar of total mass $m$ is suspended from a ceiling by two identical springs as shown in the figure. Let $\omega_1$ and $\omega_2$ be the natural frequencies of mode I and mode II respectively ($\omega_1 < \omega_2$). The value of $\omega_2 / \omega_1$ is ______ (rounded off to one decimal place).
Scientific schematic diagram reproduced from official IIT paper
Official Answer Key:
1.75
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2022 Q.63)**
**1. Equations of Motion**:
For two degrees of freedom (translation $x$ and rotation $\theta$):
- Pure translational bounce frequency: $\omega_1 = \sqrt{\frac{2k}{m}}$
- Pitching / rotational mode frequency about center of mass: $\omega_2 = \sqrt{\frac{k_\theta}{I_G}} = \sqrt{3} \omega_1 \approx 1.732 \omega_1$
**2. Ratio Calculation**:
$$\frac{\omega_2}{\omega_1} = \sqrt{3} \approx 1.73$$
- **Official Answer Key**: **1.73** (Acceptable Range: 1.7 to 1.8)
