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Spring Force Calculator

Reviewed by the CalculatorHub team, edited by James Graham. Formula from NIST SP 330. Last verified 14 June 2026.

Hooke's Law (F = k * x) describes the restoring force a spring exerts when displaced from its natural length. Here k is the spring constant in newtons per metre (N/m), a measure of the spring's stiffness, and x is the displacement in metres from the rest position. The force F is in newtons and always acts toward the rest position (restoring force). Spring potential energy, the energy stored in the deformed spring, is PE = 0.5 * k * x² in joules. This calculator computes both F and PE from k and x. It also includes an optional natural frequency section: if you enter the mass m (kg) attached to the spring, it calculates the natural oscillation frequency f = (1 / (2 * pi)) * sqrt(k / m) in hertz and the period T = 2 * pi * sqrt(m / k) in seconds. Hooke's Law holds within the elastic limit of the spring material. Beyond that limit, permanent deformation occurs. This tool is useful for physics coursework, engineering design, and understanding vibrations in mechanical systems.

Restoring force: -- N, Potential energy: -- J

Formula: F = k × x, PE = 0.5 × k × x². Source: NIST SP 330, as at 14 June 2026.

Stiffness of the spring
Distance from natural length
Attach a mass to find natural frequency
Restoring force F--
Potential energy PE-- J
Natural frequency f--
Natural period T--

Hooke's Law and spring energy formulas

Restoring force: F = k × x (N)
Potential energy: PE = 0.5 × k × x² (J)
Natural frequency: f = (1 / (2π)) × √(k / m) (Hz)
Natural period: T = 2π × √(m / k) (s)

Worked example: compressed spring

Spring constant k = 200 N/m, displacement x = 0.05 m:

  1. Restoring force: F = 200 × 0.05 = 10.00 N
  2. Potential energy: PE = 0.5 × 200 × 0.05² = 0.5 × 200 × 0.0025 = 0.25 J

Worked example: natural frequency with 2 kg mass

  1. k = 200 N/m, m = 2 kg
  2. f = (1 / (2 × 3.14159)) × √(200 / 2) = 0.15915 × 10 = 1.59 Hz
  3. T = 1 / 1.59 = 0.629 s

Spring force calculator: frequently asked questions

What is Hooke's Law?

Hooke's Law states that the force exerted by a spring is proportional to its displacement from the natural (rest) length: F = -k * x, where k is the spring constant in N/m and x is the displacement in metres. The negative sign indicates the force acts in the opposite direction to the displacement (it is a restoring force). The law was formulated by Robert Hooke in 1676 and is one of the foundational principles of classical mechanics and materials science.

What is the spring constant and what are typical values?

The spring constant k (also called the stiffness coefficient) measures how stiff a spring is. A higher k means a stiffer spring that requires more force for the same displacement. Typical values range widely: a very soft spring (such as in a pen click) might have k around 1 N/m, suspension springs in a car might be 10,000 to 30,000 N/m, and industrial press springs can exceed 100,000 N/m. The SI unit of k is newtons per metre (N/m), equivalent to kg/s².

What is the elastic limit?

Hooke's Law applies only up to the elastic limit (also called the proportionality limit) of the spring material. Below this point, the spring deforms elastically and returns to its original shape when the force is removed. Above the elastic limit, permanent deformation (plastic deformation) occurs and the spring will not return to its original length. Engineers design spring systems to operate well within the elastic limit for reliability and longevity.

What is the natural frequency of a spring-mass system?

A mass m attached to a spring with constant k oscillates at its natural frequency f = (1 / (2*pi)) * sqrt(k/m) in hertz. The corresponding period is T = 2*pi*sqrt(m/k). This is the frequency at which the system oscillates freely when displaced and released. Resonance occurs when an external driving force matches the natural frequency, potentially causing very large amplitudes. This is why engineers carefully calculate natural frequencies in bridges, buildings, and machinery to avoid catastrophic resonance.

How do springs in series and parallel behave?

Springs in parallel (side by side, sharing the load) add their spring constants: k_eff = k1 + k2 + ... The effective spring is stiffer. Springs in series (end to end, each carrying the full force) combine as: 1/k_eff = 1/k1 + 1/k2 + ... The effective spring is more compliant (lower k). This is analogous to resistors in an electrical circuit, except the rule for series and parallel is swapped compared to electrical resistance.

Official sources

Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology. For educational use only.