Hooke's Law Formula
Hooke's Law calculates the force in a spring using F = -kx.
Learn how spring constant and displacement determine restoring force with examples.
The Formula
Hooke's Law describes the restoring force exerted by a spring when it is stretched or compressed from its natural length. The negative sign indicates that the force always acts in the opposite direction of the displacement, pulling the object back toward equilibrium.
This law was discovered by English scientist Robert Hooke in 1678. It applies to any elastic material as long as the deformation stays within the elastic limit. Beyond that limit, the material deforms permanently and the law no longer holds.
The elastic potential energy stored in a spring is given by a related formula: PE = ½kx². This energy is released when the spring returns to its natural length.
Variables
| Symbol | Meaning |
|---|---|
| F | Restoring force exerted by the spring (in newtons, N) |
| k | Spring constant, a measure of stiffness (in N/m) |
| x | Displacement from the natural (rest) length (in meters, m) |
Example 1
A spring with a spring constant of 200 N/m is stretched 0.15 m from its rest position. What force does it exert?
Identify the values: k = 200 N/m, x = 0.15 m
Apply the formula: F = -kx = -(200)(0.15)
F = -30 N (the negative sign means the force pulls back toward equilibrium)
|F| = 30 N
Example 2
A 5 kg mass hangs from a spring and stretches it by 0.08 m. What is the spring constant?
At equilibrium, the spring force equals the weight: F = mg = 5 × 9.8 = 49 N
Rearrange Hooke's Law: k = F / x
k = 49 / 0.08
k = 612.5 N/m
When to Use It
Hooke's Law applies whenever you deal with springs or elastic materials within their elastic limit.
- Designing suspension systems in vehicles
- Calculating forces in spring-loaded mechanisms
- Analyzing simple harmonic motion
- Determining material stiffness in engineering applications