Hooke's Law states that the force F needed to extend or compress a spring by some distance x is proportional to that distance. Mathematically: F = −kx, where k is the spring constant (a measure of the spring's stiffness). The negative sign indicates that the restoring force always opposes the direction of displacement. This linear relationship holds as long as the material stays within its elastic limit — beyond that, permanent deformation occurs and Hooke's Law no longer applies.

In SI units, the spring constant k is measured in Newtons per meter (N/m). Other common units include:

• N/cm — Newtons per centimeter (1 N/cm = 100 N/m)
• N/mm — Newtons per millimeter (1 N/mm = 1,000 N/m)
• lb/in — Pounds per inch (common in US engineering, 1 lb/in ≈ 175.13 N/m)
• lb/ft — Pounds per foot (1 lb/ft ≈ 14.59 N/m)

A higher k value means a stiffer spring that requires more force to stretch or compress.

The elastic potential energy stored in a spring is given by: PE = ½kx²

Where:

• k = spring constant (N/m)
• x = displacement from equilibrium (m)
• PE = energy stored (Joules)

This energy is stored as a result of the work done to deform the spring. When released, this energy can be converted to kinetic energy — this principle is used in everything from pogo sticks to shock absorbers and clock mechanisms.

Springs in Series: The effective spring constant decreases. 1/k_eff = 1/k₁ + 1/k₂. Two identical springs in series have half the stiffness of a single spring.

Springs in Parallel: The effective spring constant increases. k_eff = k₁ + k₂. Two identical springs in parallel are twice as stiff.

This is analogous to electrical resistors — springs in series behave like resistors in parallel, and vice versa.

• Vehicle Suspension: Coil springs absorb road shocks; k determines ride stiffness.
• Weighing Scales: Spring scales measure weight via displacement (F = kx).
• Mechanical Watches: The mainspring stores elastic energy to power the movement.
• Shock Absorbers: Combine springs with dampers to control motion.
• Nanotechnology: AFM cantilevers behave as tiny springs measuring atomic forces.
• Medical Devices: Syringe plungers, catheter guidewires, and orthodontic braces all use spring principles.

The elastic limit is the maximum stress a material can withstand while still returning to its original shape when the load is removed. Beyond this point, the material undergoes plastic deformation — it stays permanently stretched or compressed. Hooke's Law is only valid within the elastic region. For most metals, this region is quite small (typically < 1% strain). Engineers must design spring systems to operate well within this limit to ensure reliability and safety.

This calculator uses the exact Hooke's Law formulas with double-precision floating-point arithmetic. All unit conversions follow NIST standards. Results are displayed with appropriate significant figures. The calculator handles a wide range of values — from delicate micro-springs (k ≈ 0.001 N/m) to heavy industrial springs (k > 10⁶ N/m). For educational and most engineering estimation purposes, the accuracy is more than sufficient.