What Is Hooke's Law and How Is It Calculated?
Hooke's law states that the force needed to stretch or compress a spring by some distance is directly proportional to that distance, as long as the spring stays within its elastic limit. The constant of proportionality, the spring constant k, measures how stiff the spring is, in newtons per metre (N/m).
The Hooke's law formula, published by English scientist Robert Hooke in 1678, is: F = −k x, where F is the restoring force in newtons, k is the spring constant in N/m, and x is the displacement from the spring's rest length in metres. The minus sign shows the force opposes the displacement. Rearranged, k = F ÷ x and x = F ÷ k.
The energy stored in a stretched or compressed spring is U = ½ k x² joules. Doubling the stretch therefore doubles the force but quadruples the stored energy. In the SI system a newton is defined by the General Conference on Weights and Measures as one kilogram-metre per second squared.
How to Use This Hooke's Law Calculator
Pick the quantity to solve for, then fill in the other two:
- Find Force: Enter k and the displacement x to get the spring force.
- Find k: Enter a measured force (for example a hanging weight, F = m × 9.80665) and the resulting stretch.
- Find Distance: Enter the force and k to see how far the spring stretches or compresses.
Choose metric (N, N/m, m) or US units (lbf, lbf/in, in). One lbf/in equals 175.127 N/m and one lbf equals 4.4482 N, so results are always also shown in SI units.
Typical Spring Constant Values
| Spring | k (N/m) | k (lbf/in) |
|---|---|---|
| Slinky toy | ≈ 1 – 2 | ≈ 0.006 – 0.011 |
| Ballpoint pen spring | ≈ 100 | ≈ 0.57 |
| Lab spring | ≈ 10 – 500 | ≈ 0.06 – 2.9 |
| Car suspension spring | ≈ 20,000 – 70,000 | ≈ 114 – 400 |
| Railway coil spring | ≈ 1,000,000 | ≈ 5,700 |
Hooke's law only holds up to the elastic limit. Beyond it the spring deforms permanently, and the force-extension graph is no longer a straight line.
Frequently Asked Questions About Hooke's Law
What is the formula for Hooke's law?
Hooke's law is F = −kx, where F is the restoring force in newtons, k is the spring constant in newtons per metre, and x is the displacement from the rest length in metres. The stored elastic energy is U = ½kx².
How do you find the spring constant?
Divide the applied force by the resulting displacement: k = F ÷ x. For example, a 4.905 N weight (a 0.5 kg mass) that stretches a spring by 0.098 m gives k = 50 N/m. Fitting several weights gives a more accurate slope.
What is the elastic limit of a spring?
The elastic limit is the maximum stretch or compression after which a spring returns exactly to its original length. Beyond it the spring deforms permanently and F = kx no longer applies.
How accurate is this Hooke's law calculator?
The calculator is exact for an ideal linear spring. Real springs have friction, a mass of their own, and a nonlinear range near their limits, so treat results as accurate only within the working range stated by the manufacturer.