Elastic Potential Energy in a Spring
Simple Explanation
When a spring is stretched or compressed, it stores energy — called elastic potential energy — which can later be released and converted into other forms, such as kinetic energy, as the spring returns to its natural length.
Why Do We Need It?
Elastic potential energy is a key part of understanding how energy is stored and transferred in systems like archery bows, catapults, trampolines, and mechanical clocks.
Formula
Elastic Potential Energy
E = ½kx²
The energy stored in a stretched or compressed spring, which can be recovered as the spring returns to its natural length.
- E
- — Elastic potential energy, in joules
- k
- — Spring constant, in N/m
- x
- — Extension or compression from the natural length, in metres
When to use it: Use to find the energy stored in a stretched or compressed spring, e.g. to analyse energy transfers in systems using springs.
Worked Example
Finding elastic potential energy
A spring with spring constant 300 N/m is compressed by 0.08 m. Find the elastic potential energy stored.
Why Does This Work?
Since the force needed to stretch a spring increases as it stretches (F = kx), the energy stored (work done stretching it) is not simply force × distance, but must account for this gradually increasing force — integrating this relationship produces the factor of ½ and the squared term in the energy formula.
Real-Life Example
How an archery bow launches an arrow
Drawing a bow stores energy, which is then rapidly converted into the kinetic energy of the launched arrow.
As the archer draws the bow, they do work against the bow's elastic restoring force, storing elastic potential energy — releasing the string converts this stored energy almost entirely into the arrow's kinetic energy, propelling it forward.
Practice
A spring with spring constant 500 N/m is stretched by 0.10 m. Find the elastic potential energy stored.
MediumCommon mistake
Forgetting to square the extension x, or forgetting the factor of ½ — since the restoring force increases as the spring stretches, the stored energy is NOT simply force × extension.
Quick Review
- E = ½kx² gives the elastic potential energy stored in a stretched/compressed spring.
- This energy can convert into other forms (like kinetic energy) as the spring relaxes.
- The factor of ½ and the squared x account for the increasing force as the spring stretches.