Question

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

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

Answer 1

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

The first two springs are in parallel combination
$K_{\text{eq}}=K+K=2K$


Now, the two springs are in series combination
$K_{\text{eq}}^{\prime }=\frac{\left(2K\right)\left(K\right)}{2K+K}=\frac{2K}{3}$
Hence, time period of oscillation for the block
$T=2\pi \sqrt{\frac{3m}{2K}}$