Hooke's Law Calculator

Hooke's Law describes the linear relationship between force and displacement in elastic materials like springs. The formula F = -kx states that the restoring force (F) is proportional to displacement (x) from equilibrium, with the spring constant (k) as the proportionality factor. The negative sign indicates the force always acts to restore the spring to its natural length.

The spring constant k, measured in N/m, indicates how stiff a spring is. A higher k means a stiffer spring that requires more force to stretch or compress the same distance. For example, a spring with k = 500 N/m is five times stiffer than one with k = 100 N/m. The spring constant depends on material properties and spring geometry.

Elastic potential energy is energy stored in a deformed spring, given by U = ½kx². When you stretch or compress a spring, you do work against the restoring force, and this energy is stored. When released, this energy converts to kinetic energy. The energy increases with the square of displacement - doubling the stretch quadruples the stored energy.

The elastic limit is the maximum stress a material can withstand while still returning to its original shape when the force is removed. Beyond this limit, permanent (plastic) deformation occurs, and Hooke's Law no longer applies. Springs are designed to operate well below their elastic limit to ensure repeatable behavior.

When a mass oscillates on a spring, the period T = 2π√(m/k) depends on both mass and spring constant. Heavier masses oscillate more slowly (longer period), while stiffer springs make oscillation faster (shorter period). This is simple harmonic motion - the frequency f = 1/T = (1/2π)√(k/m) is independent of amplitude.

For springs in series (end-to-end): 1/k_total = 1/k₁ + 1/k₂, making the combination softer. For springs in parallel (side-by-side): k_total = k₁ + k₂, making the combination stiffer. This is analogous to resistors in circuits but with opposite rules (series springs act like parallel resistors).

Hang known masses from the spring and measure the displacement for each. Plot force (F = mg) vs displacement (x). The slope of this line equals k. Alternatively, measure the oscillation period with a known mass: k = 4π²m/T². Multiple measurements improve accuracy.

The negative sign indicates that the force direction is opposite to the displacement direction. When you stretch a spring (positive x), it pulls back (negative F). When you compress it (negative x), it pushes out (positive F). This restoring nature is why springs oscillate when displaced and released.