Question

A body of mass $m$ hangs from three springs, each of spring constant $K$, as shown. If the mass is slightly displaced and released, the system will oscillate with time period of

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

Answer 1

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

Let $K_2$ be equivalent resistance of two springs in parallel
$K_2=K+K=2K$
As $K_2$ is in series with third spring of spring constant $K$

$K_{\text{eq}}=\frac{K\times K_2}{K+K_2}=\frac{K\times 2K}{K+2K}=\frac{2}{3}K$

Using $\text{T}=2\pi \sqrt{\frac{m}{K_{eq}}}$

$\therefore \text{T}=2\pi \sqrt{\frac{m}{\frac{2K}{3}}}$