GCSEScienceAQACombined Science: SynergyPhysical SciencesRevision NotesInteractions Over Small & Large DistancesForces & Energy ChangesEnergy Stored In A Stretched Spring

Energy Stored In A Stretched Spring (AQA GCSE Combined Science: Synergy: Physical Sciences): Revision Note

Exam code: 8465

Written by: Leander Oates

Updated on 5 March 2026

What is Elastic Potential Energy?

Energy in the elastic potential store of an object is defined as:
The energy stored in an elastic object when work is done on the object

This means that any object that can change shape by stretching, bending or compressing (eg. springs, rubber bands)
- When a spring is stretched (or compressed), work is done on the spring which results in energy being transferred to the elastic potential store of the spring
- When the spring is released, energy is transferred away from its elastic potential store

Diagram of two springs hanging from a rod, one with a weight labelled "load" causing extension, with arrows indicating force and extension. — **How to determine the extension, e, of a stretched spring**

How to Calculate Elastic Potential Energy

The amount of elastic potential energy stored in a stretched spring can be calculated using the equation:

E_e = ½ × k × e²

Where:
- E_e = elastic potential energy in joules (J)
- k = spring constant in newtons per metre (N/m)
- e = extension in metres (m)

The above elastic potential energy equation assumes that the spring has not been stretched beyond its limit of proportionality

Two springs side by side: the left spring is unstretched; the right spring has been
permanently deformed by being stretched beyond its limit of proportionality. — **The spring on the right has been stretched beyond the limit of proportionality**

Worked Example

A mass is attached to the bottom of a hanging spring with a spring constant of 250 N/m. It stretches from 10.0 cm to 11.4 cm.

Calculate the elastic energy stored by the stretched spring.

Answer:

Step 1: Determine the extension of the spring

Diagram illustrating a spring extension experiment. Initial length 10cm, final length 11.4cm, extension 1.4cm. Includes equations and a ruler for measurement.

Step 2: List the known quantities

Spring constant, k = 250 N/m
Extension, e = 1.4 cm = 0.014 m

Step 3: Write out the elastic potential energy equation

E_e = ½ ke²

Step 4: Calculate the elastic potential energy

E_e = ½ × 250 × (0.014)²

E_e= 0.0245 J

Step 5: Round the answer to 2 significant figures

E_e = 0.025 J

Examiner Tips and Tricks

Look out for units! If the question gives you units of cm for the length you MUST convert this into metres for the calculation to be correct.

To change the shape of an object, two contact forces must be acting on the object in different directions. In this case, the spring is attached to the arm of the clamp providing a contact force, and the mass is attached to the spring providing a contact force. These forces pull the spring in opposite directions and this is the reason that the spring stretches. The forces and energy topics are linked by the concept of work done which is a force acting over a distance, and also the energy transferred. Making links between concepts is an important way to fast-track your understanding.

Work Done on a Spring

When a spring is stretched or compressed by a force, work is done on the spring
Work done is the transfer of energy
- The energy is transferred to its elastic potential energy store

Three illustrations of a coil spring: static, compressed by hands with downward arrow, and stretched by hands with upward arrow. — **When a spring is stretched or compressed, there is work done and elastic potential energy is stored**

Provided the spring is not inelastically deformed (i.e has not exceeded its limit of proportionality), the work done on the spring and its elastic potential energy stored are equal

Calculating Work Done on a Spring

The work done, or the elastic potential energy stored, while stretching or compressing a spring can be calculated using the equation:

E_e = ½ × k × e²

Where:
- E_e = elastic potential energy in joules (J)
- k = spring constant in newtons per metre (N/m)
- e = extension in metres (m)

Diagram of a spring with extension "e" showing elastic potential energy formula $E_e = \frac{1}{2}Ke^2$, where K is the spring constant. — **The elastic potential energy in a stretched spring depends on its spring constant and extension**

This equation is only for springs that have not been stretched beyond their limit of proportionality
- The term e² means that if the extension is doubled then the work done is quadrupled
- This is because 2² = 4

Worked Example

A mass is attached to the bottom of a hanging spring with a spring constant k and 0.2 J of work is done to stretch it by 4.5 cm. Calculate the spring constant, k for this spring.

Answer:

Step 1: List the known quantities:
- Work done / elastic potential energy, E_e = 0.2 J
- Extension, e = 4.5 cm
Step 2: Write the relevant equation:

E_e = ½ke²

Step 3: Rearrange for the spring constant, k:

2E_e = ke²

$\frac{2 E_{e}}{e^{2}}$ = k

Step 4: Convert any units:
- Extension should be in m

4.5 cm = 0.045 m

Step 5: Substitute the values into the equation:

k = $\frac{2 E_{e}}{e^{2}}$

k = $\frac{2 \times 0.2}{{(0.045)}^{2}}$

k = 198 N/m

Examiner Tips and Tricks

Remember: when calculating the work done the extension, e, is squared (e²)!

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