The body of mass M is performing S.H.M. as shown in the figure. The periodic time ' T ' of oscillation of a given system is

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The top two springs are connected in series. Their equivalent spring constant is $K_{1} = \frac{K \cdot K}{K + K} = \frac{K}{2}$ .

The bottom two springs are connected in parallel. Their equivalent spring constant is $K_{2} = K + K = 2 K$ .

These two combinations are connected in series. The effective spring constant $K_{e q}$ of the entire system is given by:

$\frac{1}{K _{e q}} = \frac{1}{K _{1}} + \frac{1}{K _{2}}$

$\frac{1}{K _{e q}} = \frac{1}{K /2} + \frac{1}{2 K} = \frac{2}{K} + \frac{1}{2 K} = \frac{5}{2 K}$

$K_{e q} = \frac{2 K}{5}$

The time period of oscillation is $T = 2 π \frac{M}{K _{e q}}$ .

$T = 2 π \frac{M}{\frac{2 K}{5}} = 2 π \frac{5 M}{2 K}$

Answer: $2 π \frac{5 M}{2 k}$

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