Question

Find the natural frequency of oscillation of the system shown in the figure. Pulleys are massless and frictionless. Spring and strings are also massless.

• A. $\frac{\pi }{2}\text{Hz}$
• B. $\sqrt{\pi }\text{Hz}$
• C. $\frac{10}{\sqrt{\pi }}\text{Hz}$
• D. $\pi \text{Hz}$

Answer 1

The correct option is D $\pi \text{Hz}$

Let the mass $m(=1\text{kg})$ falls down by $x$, so the spring extends by $4x$.

$\therefore \frac{T}{4}=k\times 4x$

$\Rightarrow T=16kx=m{\omega }^{2}x\left[T\text{is the restoring force acting on mass}m\right]$

$\therefore \omega =\sqrt{\frac{16k}{m}}$

So,

$f=\frac{1}{2\pi }\sqrt{\frac{16k}{m}}$

$\Rightarrow f=\frac{2}{\pi }\sqrt{\frac{k}{m}}=\frac{2}{\pi }\sqrt{\frac{25}{1}}\approx \pi \text{Hz}$

Hence, option $\left(D\right)$ is the correct answer.