Spring Force
Spring force is the restoring force exerted by a compressed or stretched elastic spring, always directed back toward the spring's natural (equilibrium) length. It is a specific application of Hooke's Law and is proportional to the displacement from equilibrium. Spring forces are fundamental to mechanical oscillators, shock absorbers, force sensors, and the microscopic bonding forces between atoms.
Key Formula
F_s = −k × x
LaTeX: F_s = -kx
| Symbol | Meaning | Unit |
|---|---|---|
| F_s | Spring (restoring) force | N (Newton) |
| k | Spring constant (stiffness) | N/m |
| x | Displacement from natural length (positive = extension) | m |
Worked Example
Problem
A spring with a spring constant k = 200 N/m is stretched 0.15 m from its natural length. What is the magnitude of the spring force?
Solution
Step 1 — Identify values: k = 200 N/m, x = 0.15 m (extension). Step 2 — Apply the formula: F_s = kx = 200 × 0.15 = 30 N. The negative sign in F_s = −kx means the force acts in the direction opposite to the displacement (i.e., toward equilibrium), so the spring pulls back with 30 N.
Answer
The spring force (restoring force) has a magnitude of 30 N.
Spring force for different displacements with k = 200 N/m
| Displacement x (m) | Direction | Spring Force |F_s| (N) | Energy Stored (J) |
|---|---|---|---|
| 0.00 (natural length) | — | 0 | 0.00 |
| 0.05 (extension) | Toward equilibrium | 10 | 0.25 |
| 0.10 (extension) | Toward equilibrium | 20 | 1.00 |
| 0.15 (extension) | Toward equilibrium | 30 | 2.25 |
| 0.20 (extension) | Toward equilibrium | 40 | 4.00 |
| 0.10 (compression) | Toward equilibrium | 20 | 1.00 |
Interactive Tools
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Related Terms
Hooke's Law
Hooke's Law states that, within the elastic limit of a material, the deformation (extension or compression) of a spring or elastic solid is directly proportional to the applied force. It was formulated by English scientist Robert Hooke in 1676 and is expressed as F = kx, where k is the spring constant characterising the stiffness of the material. Hooke's Law underpins the design of force gauges, seismometers, automotive suspensions, and the theory of elasticity in materials science.
Elastic Potential Energy
Elastic potential energy is the energy stored in a deformed elastic object — such as a stretched spring, a compressed rubber band, or a bent bow — that can be fully recovered when the deforming force is removed. It arises from the intermolecular forces within the elastic material that resist deformation and is equal to the work done in stretching or compressing the object. Elastic PE is pivotal in springs, archery, vehicle suspensions, and the molecular-scale understanding of elasticity.
Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and always directed toward it. The motion follows a sinusoidal pattern over time, characterized by constant amplitude, frequency, and period in the absence of damping. SHM is the basis for understanding pendulums, springs, sound waves, and alternating electric circuits.
The word "spring" derives from Old English "springan" (to leap, burst forth), reflecting the elastic rebound. "Force" comes from Latin "fortia" (strength, power). The systematic study of spring elasticity was formalised by Robert Hooke in 1678.