Hooke’s Law and Spring Force

Hooke’s law states that the force needed to stretch or compress a spring is proportional to its extension, F = kx, where k is the spring constant in newtons per meter (N/m) and x is the change in length from the natural length. In brief: the restoring force of the spring is F = −kx, pointing opposite to the displacement. The law holds up to the limit of proportionality; beyond the elastic limit the spring is permanently deformed. The energy stored is E = ½kx².

Spring balances, car suspensions, trampolines and mattresses all depend on elastic forces. This guide states Hooke’s law and explains the spring constant and the limits of the law, describes how to find k experimentally, compares stiff and soft springs in a table, works through a calculation including stored energy and corrects common mistakes.

What is Hooke’s law?

For a spring, the applied force F is proportional to the extension x: F = kx. The spring constant k measures the stiffness: a large k means a stiff spring. The unit is N/m. The force exerted by the spring on whatever stretches it is the restoring force, F = −kx. The minus sign shows that it points back toward the natural length: if the spring is stretched (x positive), the force pulls back; if compressed (x negative), the force pushes outward. The graph of force against extension is a straight line through the origin, and its gradient is k.

Two springs can be combined: in parallel (side by side) the constants add, k = k₁ + k₂, and in series (end to end) 1/k = 1/k₁ + 1/k₂.

What are the limits of the law?

• Limit of proportionality: beyond this extension, the force-extension graph curves, and Hooke’s law no longer applies.
• Elastic limit: up to this point, the spring returns to its original length when the load is removed. Beyond it, there is a permanent deformation.
• Other elastic materials, such as a metal wire, follow the law for small strains, but rubber bands do not.

How can k be measured?

Hang a spring from a stand and measure the natural length. Add masses (known weights) one at a time, and measure the new length each time. The extension is the new length minus the natural length. Plot force (weight = mg) against extension: the slope of the line of best fit is k. Use a pointer and a ruler fixed next to the spring, and read at eye level.

Spring constants compared

Worked example: a spring with a hanging load

In a school experiment, loads are hung from a spring and the extension is measured.

1. Check proportionality: the force ÷ extension is the same for all values: 2.0 ÷ 0.010 = 200 N/m, 4.0 ÷ 0.020 = 200 N/m, 6.0 ÷ 0.030 = 200 N/m. So the spring obeys Hooke’s law with k = 200 N/m (extensions converted to meters).
2. Force for a 0.15 m extension: F = kx = 200 × 0.15 = 30 N.
3. Extension with a 0.50 kg mass: the weight is 0.50 × 9.8 = 4.9 N, so x = F ÷ k = 4.9 ÷ 200 = 0.0245 m = 2.5 cm.
4. Energy stored at 0.15 m: E = ½kx² = ½ × 200 × 0.15² = 100 × 0.0225 = 2.25 J.

How was the law discovered?

Robert Hooke found the law in 1660 and published it in 1678 as a Latin anagram, “ceiiinosssttuv”, whose solution is ut tensio, sic vis (“as the extension, so the force”). He was interested in springs for watches. The law is the basis of the theory of elasticity, and Thomas Young defined the modulus for materials in 1807.

Which mistakes are common?

Mistake 1: Using the total length instead of the extension

x is the change in length from the natural length, not the length of the spring.

Mistake 2: Forgetting to convert centimeters to meters

k is in N/m. Use extensions in meters.

Mistake 3: Applying the law beyond the limit

Beyond the limit of proportionality, force is no longer proportional to extension.

Mistake 4: Confusing the force on the spring with the force by the spring

They have the same size and opposite directions (Newton’s third law).