Hooke's law: within a spring's elastic limit, the restoring force is proportional to how far it is stretched — F = k·x, with the stored elastic energy ½kx². This free calculator solves for the force, the spring constant or the extension, in any unit, and shows every step of the working.
Stretch or compress a spring and it pushes back. Within its elastic limit, that restoring force is proportional to how far the spring has moved from its natural length: F = k·x. Here k is the spring constant, the stiffness of the spring measured in newtons per metre, and x is the extension. The force points back toward equilibrium, which is why Hooke's law is often written F = −kx — the minus sign records the direction, while the size of the force is simply kx.
There are three steps. First, decide what you want — the force, the spring constant, or the extension — and pick it in the calculator's Solve for menu. Second, enter the values you know: the force in newtons, the spring constant in newtons per metre (or N/cm), and the extension in metres, centimetres or millimetres. Third, read the answer with the worked steps, which show the formula, your numbers substituted in, and the result in its units. Every answer also reports the stored elastic energy, ½kx², in joules.
Two relationships are worth feeling directly. Force is proportional to extension, so a graph of force against extension is a straight line through the origin whose slope is the spring constant — double the stretch and you double the force. The stored energy, by contrast, grows with the square of the extension: it is the area under that force–extension line, a triangle of area ½kx², so stretching a spring twice as far stores four times the energy. A stiff spring (large k) needs a big force for a small extension; a soft one (small k) stretches easily.
Hooke's law is the gateway to two further topics. The constant itself is the subject of the spring constant calculator, k = F/x, and because the restoring force is linear in displacement, a mass on a spring oscillates with simple harmonic motion. To explore the elastic limit and force–extension behaviour in more depth, see our guide to Hooke's law, or look up a term in the physics glossary.
A spring has a spring constant k = 200 N/m and is stretched by x = 0.10 m. The restoring force is F = k·x = 200 × 0.10 = 20 N, and the stored elastic energy is PE = ½·k·x² = ½ × 200 × 0.10² = 1.0 J. Stretch it twice as far, to 0.20 m, and the force doubles to 40 N — but the stored energy quadruples to 4.0 J, a direct illustration of the linear force and the squared-energy relationship in Hooke's law.
Hooke's law underpins springs and vehicle suspensions, force sensors and spring scales, the calibration of weighing equipment, and materials testing of stiffness and elasticity. Because the restoring force is linear in displacement, it is also the foundation of simple harmonic motion — from pendulum clocks and tuning forks to the modelling of atomic bonds as tiny springs.
Hooke's law states that, within a spring's elastic limit, the restoring force is proportional to the extension: F = kx, where k is the spring constant (stiffness) and x is how far the spring is stretched or compressed from its natural length. The force points back toward equilibrium, which is why the law is often written F = −kx — the minus sign marks the direction, while the magnitude is simply kx.
Force F is in newtons (or kilonewtons), the spring constant k is in newtons per metre (or N/cm), and the extension x is in metres, centimetres or millimetres. The calculator converts every input to base SI units before solving, then reports the result and the stored elastic energy ½kx² in joules.
The spring constant k = F/x is the stiffness of the spring: the force needed to stretch it by one metre. A high k (a stiff spring) needs a large force for a small extension; a low k (a soft spring) stretches easily. It is the constant of proportionality in Hooke's law and is measured in newtons per metre.
The elastic potential energy stored in a spring obeying Hooke's law is PE = ½kx². It equals the area under the force–extension graph, which is a triangle of height kx and base x. Because the energy depends on x², stretching a spring twice as far stores four times the energy. The calculator shows this stored energy alongside every answer.
Hooke's law holds only up to the elastic limit. Within that range the spring returns to its original length when released and force stays proportional to extension. Beyond the limit of proportionality the force–extension line curves, and past the elastic limit the material yields and deforms permanently — so the simple F = kx relationship no longer applies.