Please see diagram.
The pendulums are each displaced by an amount x. The string is stretched by 2x totally. The restoration force in the spring = F2 = k * 2x = 2k x. (Hooke's law).
Bob's displacement X in the tangential direction to the string = X
x = displacement in the horizontal direction = X Cos Ф
x = L sin Ф
mg CosФ = Tension T in the string
F1 = component of weight along the tangential direction
F1 = mg SinФ = m g x / L
Net force on the bob = F = m a = - m * d²X / dt²
(-ve sign is because the force is in the decreasing x direction)
X = x / cos Ф = L tan Ф ≈ L Sin Ф ≈ L Ф, approximation for small Ф.
Net force on pendulum in the tangential direction =
F1 and F2 are at an angle Ф, and their vector sum is to be calculated for the resultant force and acceleration. But since Ф is small, we approximate the resultant force to be along horizontal direction and F1 and F2 along this direction. So we take the linear sum for simplicity.
a = - d² X / dt² ≈ - d² x / dt²
=> - m d² x / dt² = 2k x + m g x/L
Resultant force on the bob = mg x/L + 2 k x = F = m a = - m d^2 x/dt^2 d² x/dt² = - (g/L + 2k/m) x
The equation of motion above indicates that for small amplitudes x and displacements, the pendulum oscillates in a SHM and the corresponding Angular velocity = ω
= √(g/L + 2k/m)= Time period
If displacements are large then, the resultant force F will be: