Question

A block of mass $m$ is arranged with massless pulleys and spring as shown in figure. The natural frequency of oscillation of the system is

• A. $\frac{2}{\pi }\sqrt{\frac{2k}{m}}$
• B. $2\pi \sqrt{\frac{3m}{8k}}$
• C. $\frac{1}{\pi }\sqrt{\frac{k}{m}}$
• D. $\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

Answer 1

The correct option is B $2\pi \sqrt{\frac{3m}{8k}}$

Draw a diagram for the given question and make change as per the need.
Let mass $m$ falls down by $x$ so spring extends by $4x$.
$\frac{T}{4}=K\left(4x\right)$
$T4=16kx$
Hence, $(K=16k)$
Here, $T$ is the restoring force on mass $m$.
The natural frequency of oscillation of the system is,
$f=\frac{1}{2\pi }\sqrt{\frac{K}{m}}$
$f=\frac{1}{2\pi }\sqrt{\frac{16k}{m}}$
$f=\frac{4}{2\pi }\sqrt{\frac{k}{m}}$
$f=\frac{2}{\pi }\sqrt{\frac{k}{m}}$
Final answer : (d)