What it does
A mass hanging on a spring sinks until the spring’s pull balances its weight, then bobs up and down about that resting point if you pull it and let go. This page shows that with one spring, with two springs in series (one below the other) and with two springs in parallel (side by side, holding a bar). It works out the combined spring constant, how far the springs stretch at rest (x = mg/k), the period T = 2π√(m/k) and the energy of the bobbing, with your numbers in every formula.
Joining springs is a common exam question because the answers feel backwards at first: two springs in series are softer than either one, and two in parallel are stiffer. The picture makes it clear: in series each spring carries the whole weight and the stretches add up; in parallel they share the weight and stretch together.
It serves Class 11 Physics, Oscillations (the spring-mass system, springs in series and parallel, the period of a loaded spring) and Mechanical Properties of Solids (Hooke’s law), and the Class 11 practical on finding the spring constant from its stretch and from its period. See SHM circle for why the motion is simple harmonic, pendulum lab for the other classic oscillator, and coupled oscillators for several masses joined by springs.
How to use it
- Choose One spring, Series or Parallel.
- Set the spring constants k₁ (and k₂): how many newtons it takes to stretch each spring by one metre.
- Set the mass and how far you pull it down before letting go (the amplitude).
- The dashed lines mark the spring’s natural length and the rest position, where the weight is balanced. The picture’s stretch is scaled to fit.
- Series or parallel? compares both ways of joining the same two springs.
Worked example
As the page opens: one spring with k = 20 N/m and a 0.5 kg mass. T = 2π√(0.5 / 20) = 0.9935 s, and at rest the spring is stretched x = 0.5 × 9.8 / 20 = 24.5 cm. Pulled down 5 cm, it bobs with energy ½ × 20 × 0.05² = 0.025 J and a top speed of 31.6 cm/s.
Two such springs in series: 1/k = 1/20 + 1/20, so k = 10 N/m and T = 1.405 s. In parallel: k = 40 N/m and T = 0.7025 s. The series pair is exactly 2 times slower than the parallel pair.
Check yourself
Five fresh questions: a series pair, a parallel pair, a period, a resting stretch, and a spring cut into pieces. Answers are checked on the page and the working is shown. Nothing is saved.
Tips
- For two springs in series, k = k₁k₂ / (k₁ + k₂): “product over sum”, like resistors in parallel.
- T does not depend on g or on the amplitude, only on m and k. The same spring bobs at the same rate on the Moon, although it hangs less stretched.
- Because x = mg/k, the period can also be written T = 2π√(x/g): measure how far a mass stretches a spring and you can predict how fast it bobs.
FAQ
Why do series springs get softer? The same force runs through both springs, so each stretches by F/kᵢ. The total stretch is F/k₁ + F/k₂, which is more than either alone, so the pair has a smaller k.
Why is a cut spring stiffer? Half a spring stretches half as much under the same force (it is one of two equal springs in series), so its k is doubled.
What about the mass of the spring? Real springs have mass, which makes the period a little longer (roughly as if a third of the spring’s mass were added to the load). This page treats the springs as light.
Where the numbers come from
- Hooke's law F = kx, springs in series 1/k = 1/k₁ + 1/k₂ and in parallel k = k₁ + k₂, period of a mass on a light spring T = 2π√(m/k), static stretch x = mg/k, energy ½kA²: standard mechanics.
- g = 9.8 m/s² (school value).
- Checked against an independent Python calculation and a step-by-step simulation of two springs in series.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.