Hooke's Law (Spring Constant)
Hooke's law F = -kx describes the restoring force of a spring.
Calculate spring constant, force, and displacement.
The Formula
Hooke's law states that the force exerted by a spring is proportional to its displacement from the equilibrium position. The negative sign indicates the force acts in the opposite direction of the displacement (a restoring force).
Variables
| Symbol | Meaning |
|---|---|
| F | Restoring force exerted by the spring (measured in Newtons, N) |
| k | Spring constant (measured in Newtons per meter, N/m) — stiffness of the spring |
| x | Displacement from equilibrium position (measured in meters, m) |
Elastic Potential Energy
The energy stored in a stretched or compressed spring is:
This energy can be converted to kinetic energy when the spring is released. This is the principle behind catapults, trampolines, and mechanical watches.
Example 1
A spring with k = 200 N/m is stretched 0.15 m from its natural length. What force does it exert?
Apply Hooke's law: F = -kx = -(200)(0.15)
F = -30 N (the negative sign means the force pulls back toward equilibrium)
Example 2
A force of 50 N stretches a spring by 0.25 m. What is the spring constant?
Rearrange: k = F / x = 50 / 0.25
k = 200 N/m
Example 3
How much energy is stored in a spring (k = 500 N/m) compressed by 0.10 m?
Apply the elastic PE formula: PE = ½kx² = ½(500)(0.10)²
PE = ½(500)(0.01) = 250 × 0.01
PE = 2.5 Joules
When to Use It
Hooke's law applies to any elastic material within its proportional limit.
- Calculating the force required to stretch or compress a spring by a specific amount
- Determining the spring constant from force and displacement measurements
- Calculating energy stored in springs, bungee cords, and elastic bands
- Simple harmonic motion problems (mass on a spring)
- Engineering applications: suspension systems, spring scales, shock absorbers
- Note: Hooke's law only applies within the elastic limit. Beyond that, the material deforms permanently.