Five identical springs are used in the three configurations as...

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.

Options

A $1: \sqrt{2}: \frac{1}{\sqrt{2}}$
B $2: \sqrt{2}: \frac{1}{\sqrt{2}}$
C $\frac{1}{\sqrt{2}}: 2: 1$
D $2: \frac{1}{\sqrt{2}}: 1$

Correct answer

A. $1: \sqrt{2}: \frac{1}{\sqrt{2}}$

Step-by-step solution

\begin{aligned} & \quad T_a=2 \pi \sqrt{\frac{m}{k}} \\ & T_b=2 \pi \sqrt{\frac{m}{(k / 2)}} \\ & T_c=2 \pi \sqrt{\frac{m}{2 k}} \\ & \therefore \quad T_a: T_b: T_c=1: \sqrt{2}: \frac{1}{\sqrt{2}} \end{aligned}

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