recall and use the formula for the spring constant k = F / x

Stress & Strain – The Spring Constant

What is the Spring Constant?

The spring constant, usually denoted by k, tells us how stiff a spring is.

It is defined by Hooke’s Law:

\$k = \frac{F}{x}\$

Where F is the force applied (in newtons, N) and x is the resulting extension or compression (in metres, m). The larger the value of k, the stiffer the spring.

Exam Tip

  • Always write k in N m⁻¹ (or kg s⁻²).

  • Check the sign of F – for compression, F is still positive; x is taken as a positive magnitude.

  • When given F and x, compute k by dividing.

  • When given k and x, find F = k x.

  • When given k and F, find x = F / k.

Analogy: The Rubber Band

Imagine a rubber band stretched between your fingers.

If you pull it a little, it resists with a small force. Pull it twice as far, and the force roughly doubles. That’s Hooke’s Law in action – the rubber band’s “spring constant” is the ratio of force to stretch.

Typical Spring Constants

Spring TypeTypical k (N m⁻¹)
Small desk spring10–50
Car suspension spring2000–5000
Heavy-duty industrial spring10⁵–10⁶

Example: Measuring a Spring Scale

  1. Hang a known weight (e.g., 5 kg) from a spring scale. The scale reads the force F (≈ 49 N).

  2. Measure the extension x of the spring (say 0.02 m).

  3. Calculate the spring constant: k = F / x = 49 N / 0.02 m = 2450 N m⁻¹.

  4. Check that the spring behaves linearly by repeating with different weights.

Graphical Method

Plot F (y‑axis) against x (x‑axis).

The slope of the straight‑line portion of the graph is the spring constant k:

\$k = \frac{\Delta F}{\Delta x}\$

Potential Energy Stored in a Spring

When a spring is stretched or compressed, it stores elastic potential energy:

\$U = \tfrac{1}{2} k x^2\$

This is useful for calculating the energy released in a spring‑powered toy or the work done by a spring in a machine.

Quick Formula Recap

  • Hooke’s Law: F = kx

  • Spring Constant: k = F / x

  • Extension: x = F / k

  • Potential Energy: U = ½ k x²

Remember!

The spring constant is only valid within the elastic limit – the range where the spring returns to its original shape after the force is removed. Beyond this, the spring may permanently deform or break.