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. $\frac{1}{\pi} \sqrt{\frac{2 k}{m}}$

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
$\Rightarrow \frac{T}{4} = k(4x)$
$T = (16 k)x$
Thus equation of motion of mass $m$ is
$T = ma$
$\Rightarrow a = - \frac{16 k}{m} x$
omparing with $a = -\omega^2 x$
we get $\omega = \sqrt{\frac{16 k}{m}}$
$\Rightarrow f = \frac{\omega }{2\pi} = \frac{1}{2 \pi} \sqrt{16 k}{m}$
$ = \frac{2}{\pi} \sqrt{k}{m}$