Question

Find the natural frequency of oscillation of the system as shown in figure. Pulleys are massless and frictionless. Spring and string are also massless. (Take ${\pi }^2=10)$

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

Answer 1

The correct option is D $\pi$ $Hz$

With careful analysis we can see that if mass $m$ falls down by $x$ distance, the spring extends by $4x$;
$\therefore \frac{T}{4}=k\left(4x\right)$
$T=\left(16k\right)x$
where $T$ is the restoring force on mass $m$
$\therefore f=\frac{1}{2\pi }\sqrt{\frac{16k}{m}}$
$\Rightarrow f=\frac{2}{\pi }\sqrt{\frac{k}{m}}=\frac{2}{\pi }\times \frac{5}{1}=\frac{{\pi }^2}{\pi }=\pi Hz$