Hooke's Law
Hooke's law F = -kx describes the restoring force of a spring.
Learn to calculate spring force with the spring constant and displacement.
The Formula
The force exerted by a spring is proportional to its displacement from the natural length. The negative sign indicates the force acts in the opposite direction to the displacement (restoring force).
Variables
| Symbol | Meaning |
|---|---|
| F | Restoring force of the spring (measured in newtons, N) |
| k | Spring constant (measured in newtons per meter, N/m) |
| x | Displacement from the natural (unstretched) length (measured in meters, m) |
Example 1
A spring with a spring constant of 250 N/m is stretched 0.08 m from its natural length. What force does the spring exert?
Identify the values: k = 250 N/m, x = 0.08 m
Apply the formula: F = kx = 250 × 0.08 (using magnitude)
F = 20 N (directed back toward the natural length)
Example 2
A spring requires 45 N of force to stretch it by 0.15 m. What is the spring constant?
Rearrange: k = F / x
k = 45 / 0.15
k = 300 N/m
When to Use It
Use Hooke's law for problems involving elastic materials and springs.
- Calculating the force needed to stretch or compress a spring
- Determining the spring constant from measurements
- Designing suspension systems, scales, and shock absorbers
- Only valid within the elastic limit (before permanent deformation)
Key Notes
- Restoring force direction: The negative sign in F = −kx is crucial — it means the spring force always acts opposite to the displacement, pulling the object back toward equilibrium.
- Elastic limit: Hooke's law is only valid within the elastic limit. Beyond this point, the material permanently deforms and no longer returns to its original shape when the force is removed.
- Springs in series and parallel: Springs in series: 1/k_total = 1/k₁ + 1/k₂ (effective constant decreases). Springs in parallel: k_total = k₁ + k₂ (effective constant increases). Series springs are softer; parallel springs are stiffer.
- Leads to simple harmonic motion: When a mass on a spring is displaced and released, it oscillates with angular frequency ω = √(k/m) and period T = 2π√(m/k). This is the basis of clock pendulums and vibration isolation systems.
- Beyond springs: Hooke's law applies broadly to elastic deformation — rubber bands, bungee cords, and even atomic bonds behave according to this law within their elastic range.