Question

Five identical springs are used in the three configurations as shown in figure. The time periods of vertical oscillations in configurations (a), (b) and (c) are in the ratio:
\includegraphics[width=0.5\linewidth]{PH39.png}

• A. \( 1:\sqrt{2}:\frac{1}{\sqrt{2}} \)
• B. \( 2:\sqrt{2}:\frac{1}{\sqrt{2}} \)
• C. \( \frac{1}{\sqrt{2}}:2:1 \)
• D. \( 2:\frac{1}{\sqrt{2}}:1 \)

Hint:

For springs in series, the effective spring constant is given by \( \frac{1}{k_{eff}} = \sum \frac{1}{k_i} \). For springs in parallel, the effective spring constant is given by \( k_{eff} = \sum k_i \). The time period of a spring-mass system is \( T = 2\pi \sqrt{\frac{m}{k}} \).