Position 2: spring stretched by δ due to weight W force and is in equilibrium
W=kδ ---------------(1)
Position 3: spring stretched further by x at time t when it is having a simple harmonic motion about point o
Using Netwons equation of motion
ΣF = ma
W-k(δ+x)=W/g*
W-kδ-kx=W/g*
But from equation 1 W=kδ
W-W-kx=W/g*
W/g*
+(kg/W)*x ---------------------------(2)
EQ 2 resembles simple harmonic motion which is given by the EQ
+ω
2x =0 -----------(3)
From EQ 2 & 3
ω2 = kg/W
ω=√kg/W
ω=√k/m ( since W=mg)
but t=2π/ω
t=2π√m/k
Frequency 1/t =fn = 1/2π * √k/m