the final time period will be root 2 times the initial time period that is

T- = root 2 T

for solution use t = 2 pie (m/k)^1/2

initially T = 2 pie (M/K)^1/2

now, final time period = 2 pie (2M/k)^1/2

as now the net deforming force is 2Mg = k x-

Tf = (2)^1/2 2 pie (M/k)^1/2

Tf = 2^1/2 Ti = 2^1/2 T