Question

A uniform rigid rod of mass m = 1 kg and length L = 1 m is hinged at its centre & laterally supported at one end by a spring of spring constant k = 300 N/m. The natural frequency ${\omega }_n$ in rad/s

• A. 10
• B. 20
• C. 30
• D. 40

Answer 1

The correct option is C 30


$\frac{1}{2}I\widehat{{\theta }^2}+\frac{1}{2}{x}^2=\text{constant}$

$\frac{1}{2}I\widehat{{\theta }^2}+\frac{1}{2}k(\frac{L}{2}\theta {)}^2=\text{constant}$

Differentiating the whole expression w.r.to t

$\frac{1}{2}I2\dot{\theta }\ddot{\theta }+\frac{k}{2}\times \frac{L^2}{4}\times 2\theta \dot{\theta }=0$

$\Rightarrow I\ddot{\theta }+\frac{K{L}^2}{4}\theta =0$

Putting the value of $I=\frac{m{L}^2}{12}$

$\Rightarrow \frac{m{L}^2}{12}\ddot{\theta }+\frac{k{L}^2}{4}\dot{\theta }=0\Rightarrow \ddot{\theta }+\frac{3k}{m}\theta =0$

$\therefore \text{Natural frequency}:$

${\omega }_n=\sqrt{\frac{3k}{m}}$

$=\sqrt{\frac{3\times 300}{1}}=30\text{rad/s}$