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.

• A. $\frac{1}{7\pi }\sqrt{\frac{k}{m}}$
• B. $\frac{1}{2\pi }\sqrt{\frac{k}{m}}$
• C. $\frac{2}{\pi }\sqrt{\frac{k}{m}}$
• D. $\frac{1}{\pi }\sqrt{\frac{2k}{m}}$

Answer 1

Answer: (C)

Let mass '$m$' falls down by $x$ so spring extends by $4x$ ;
which causes an extra tension $T$ in lowest string
$\Rightarrow \frac{T}{4}=k(4x)$
$T=(16k)x$
Thus equation of motion of mass $m$ is
$T=ma$
$\Rightarrow a=-\frac{16k}{m}x$
omparing with $a=-{\omega }^{2}x$
we get $\omega =\sqrt{\frac{16k}{m}}$
$\Rightarrow f=\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{16k}m$
$=\frac{2}{\pi }\sqrt{k}m$