Hooke's Law and the Spring Constant
Simple Explanation
Hooke's law states that, within its elastic limit, the force needed to stretch or compress a spring is directly proportional to the extension or compression — doubling the extension doubles the force needed. The spring constant (k) measures how stiff the spring is.
Why Do We Need It?
Hooke's law is fundamental to understanding springs, and by extension, many other elastic materials and structures that behave similarly within certain limits — from car suspensions to weighing scales.
Formula
Hooke's Law
F = kx
Within its elastic limit, the force needed to stretch or compress a spring is directly proportional to how far it is stretched or compressed.
- F
- — Applied force (or restoring force), in newtons
- k
- — Spring constant — a measure of the spring's stiffness, in N/m
- x
- — Extension or compression from the natural (unstretched) length, in metres
When to use it: Use to find the force needed to stretch a spring by a known amount, or the extension produced by a known force — valid only within the spring's elastic limit.
Worked Example
Finding a spring's extension
A spring has a spring constant of 250 N/m. Find its extension when a 15 N force is applied.
Why Does This Work?
Within the elastic limit, a spring's molecular structure stretches in a way that produces a restoring force directly proportional to the deformation — beyond the elastic limit, this proportionality breaks down and the spring may be permanently deformed.
Real-Life Example
Spring weighing scales
A simple spring scale measures weight by how far a spring stretches when an object is hung from it.
Because extension is directly proportional to force (weight) via Hooke's law, the scale can be calibrated with a simple linear scale — twice the weight produces exactly twice the extension.
Practice
A spring with spring constant 400 N/m is stretched by 0.05 m. Find the force required.
MediumCommon mistake
Applying Hooke's law beyond the spring's elastic limit — past this point, the relationship between force and extension is no longer proportional, and the spring may not return to its original shape.
Quick Review
- F = kx: force is directly proportional to extension, within the elastic limit.
- k (spring constant) measures stiffness — a larger k means a stiffer spring.
- Beyond the elastic limit, Hooke's law no longer applies.