Springs in Series and Parallel Calculator
Combine up to three spring constants in series or parallel to get the equivalent stiffness, total extension under a load, force and stretch of each spring, stored energy and oscillation period.
How it works
Series: 1/k = 1/k₁ + 1/k₂ + 1/k₃ (same force, extensions add) · Parallel: k = k₁ + k₂ + k₃ (same extension, forces add) · x = F/k · E = F²/(2k) · T = 2π√(m/k)
Springs in series carry the same force, so their extensions add and the combination is softer than the softest spring: the reciprocals of the stiffnesses add. Springs side by side in parallel share the same extension, so their forces add and the stiffnesses simply sum. These are the mirror image of the rules for resistors (or the same as for capacitors). Each spring is assumed to obey Hooke’s law F = kx and to be massless, as in OpenStax University Physics.
Worked example
Two springs of 200 N/m and 300 N/m in series: 1/k = 1/200 + 1/300, so k = 120.00 N/m. A 10 N load stretches the pair x = 10 / 120 = 8.333 cm — 5.000 cm in the first spring and 3.333 cm in the second, each carrying the full 10 N. The stored energy is 10² / (2 × 120) = 0.4167 J, and a 1 kg mass would oscillate with T = 2π√(1 / 120) = 0.5736 s (1.743 Hz).
Sources: OpenStax University Physics Vol. 1 — Simple Harmonic Motion (springs) · Wikipedia — Series and parallel springs
Frequently asked questions
Are springs in series stiffer or softer?
Softer. The equivalent constant is always below the smallest individual constant — two identical springs in series have half the stiffness of one.
Why is the rule the opposite of resistors?
For resistors in series the voltage drops add; for springs in series the extensions add, which corresponds to adding compliances 1/k. Springs therefore behave like capacitors: series uses reciprocals, parallel adds directly.
What if the parallel springs are not attached symmetrically?
The parallel rule assumes the load bar stays level so every spring stretches the same amount. If the load is off-centre the bar tilts and each spring stretches differently.
How do I cut a spring in half?
A spring is a series chain of its own coils, so half the length has twice the spring constant (k ∝ 1/number of active coils).
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Results assume the idealised textbook model stated above; real systems have losses and tolerances, so verify safety-critical values with measurements or a qualified engineer.