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:..