Question

A block of mass $m$ is connected to springs as shown in the figure. Find the time period of small oscillations.
[All springs are identical]

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

Answer 1

The correct option is A $T=2\pi \sqrt{\frac{m}{3K}}$


Springs $\left(1\right)$ and $\left(2\right)$ are in parallel combination
Thus, $K_{\text{eq}}^{\prime }=K_1+K_2=2K$
Now, spring $\left(3\right)$ and combination of spring (1) and $\left(2\right)$ are in parallel again.
$\therefore K_{\text{eq}}=K_3+K_{eq}^{\prime }=K+2K=3K$
Time period of oscillation is given by
$T=2\pi \sqrt{\frac{m}{K_{\text{eq}}}}$
$\Rightarrow T=2\pi \sqrt{\frac{m}{3K}}$
Thus, option (a) is the correct answer.