Springs in Series

When two springs are placed in series the total extension is the sum of the individual extensions of each spring.

ΔLP	=	W kP
ΔLQ	=	W kQ

ΔLP + ΔLQ

=

W kP

+

W kQ

Effective spring constant

1 k

=

1 kP

+

1 kQ

In series, the same force W acts through each spring, so each spring stretches as if it carried the whole load on its own. The extension of each spring is the weight divided by its own spring constant, and the total extension is the sum.Rearranging 1/k = 1/k_P + 1/k_Q gives the effective spring constant as the product over the sum:k = k_Pk_Q / (k_P + k_Q)Two identical springs in series give half the stiffness of one spring, so the extension is doubled. Springs in series are always less stiff than either spring on its own.Worked example: springs of 2000 N m^-1 and 2500 N m^-1 in series carry a load of 1850 N. The extensions are 1850/2000 = 0.925 m and 1850/2500 = 0.740 m, giving a total of 1.665 m. Check: k = (2000 × 2500)/4500 = 1111 N m^-1, and 1850/1111 = 1.665 m.