Initially system is in equilibrium. Time period of $SHM$ of block in vertical direction is
Diffcult
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Net vertical force
$=\mathrm{kx}+\frac{\mathrm{kx}}{\sqrt{2}} \cos 45 \times 2=2 \mathrm{kx}=\mathrm{ma}$
$\mathrm{a}=\frac{2 \mathrm{k}}{\mathrm{m}} \mathrm{x}$
$\Rightarrow \omega=\sqrt{\frac{2 \mathrm{k}}{\mathrm{m}}}$
$\tau=2 \pi \sqrt{\frac{\mathrm{m}}{2 \mathrm{k}}}$
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