The body of mass M is performing S.H.M. as shown in the figure. The...

The top two springs are connected in series. Their equivalent spring constant is $K_1 = \dfrac{K \cdot K}{K + K} = \dfrac{K}{2}$.

The bottom two springs are connected in parallel. Their equivalent spring constant is $K_2 = K + K = 2K$.

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

$\dfrac{1}{K_{eq}} = \dfrac{1}{K_1} + \dfrac{1}{K_2}$

$\dfrac{1}{K_{eq}} = \dfrac{1}{K/2} + \dfrac{1}{2K} = \dfrac{2}{K} + \dfrac{1}{2K} = \dfrac{5}{2K}$

$K_{eq} = \dfrac{2K}{5}$

The time period of oscillation is $T = 2\pi\sqrt{\dfrac{M}{K_{eq}}}$.

$T = 2\pi\sqrt{\dfrac{M}{\dfrac{2K}{5}}} = 2\pi\sqrt{\dfrac{5M}{2K}}$

Answer: $2\pi\sqrt{\dfrac{5M}{2k}}$