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:

..