Elastic Potential Energy

Pull back the elastic of a catapult, wind up a clockwork toy, draw a bow, or compress the spring in a pinball launcher, and you store energy ready to be released in an instant. This is elastic potential energy — the energy stored in an object when it is stretched or compressed. The moment you let go, that stored energy is transferred, usually to the kinetic store of whatever the spring pushes or pulls. This lesson, part of Topic P7 (Energy) of OCR Gateway Science A, defines elastic potential energy, introduces and rearranges the equation $E_e = \dfrac{1}{2}ke^2$ , links it to Hooke's law from Topic P2, and works through calculations for the energy stored in springs.

By the end of this lesson you should be able to define elastic potential energy, use and rearrange $E_e = \dfrac{1}{2}ke^2$ , link it to Hooke's law and the spring constant, and calculate the energy stored in a stretched or compressed spring.

What Is Elastic Potential Energy?

Elastic potential energy is the energy stored in an elastic object (such as a spring, an elastic band or a bungee cord) when it is stretched or compressed. Doing work to deform the object — pulling it longer or squashing it shorter — transfers energy to its elastic potential store. When the object is released and returns to its original shape, that stored energy is transferred away again, usually becoming kinetic energy.

For this store to behave simply, the object must be elastic — it must return to its original shape when the force is removed. A spring stretched within its limits springs back; a piece of modelling clay does not (it stays squashed), so it does not store elastic potential energy in the same way. Elastic potential energy, like all energy, is measured in joules (J).

Exam Tip: Elastic potential energy is stored in a stretched or compressed elastic object. It applies to both stretching and compressing — a squashed spring stores elastic potential energy just as a stretched one does.

Linking to Hooke's Law

In Topic P2 (Forces) you met Hooke's law: for an elastic object such as a spring, the extension is proportional to the force, provided the limit of proportionality is not exceeded. Hooke's law is written:

$F = ke$

where $F$ is the force (in newtons, $\text{N}$ ), $k$ is the spring constant (in newtons per metre, $\text{N/m}$ ) and $e$ is the extension (in metres, $\text{m}$ ). The spring constant $k$ is a measure of the stiffness of the spring — a stiff spring has a large $k$ and needs a big force to stretch it, while a floppy spring has a small $k$ .

The work you do stretching the spring is stored as elastic potential energy. The equation for this stored energy is closely related to Hooke's law, and uses the same spring constant $k$ and same extension $e$ .

Exam Tip: The spring constant $k$ (in $\text{N/m}$ ) measures stiffness — a stiffer spring has a larger $k$ . The same $k$ appears in both Hooke's law ( $F = ke$ ) and the elastic potential energy equation ( $E_e = \dfrac{1}{2}ke^2$ ).

The Elastic Potential Energy Equation

The elastic potential energy stored in a stretched or compressed spring is:

$E_e = \dfrac{1}{2}ke^2$

where $E_e$ is the elastic potential energy (in joules, $\text{J}$ ), $k$ is the spring constant (in newtons per metre, $\text{N/m}$ ) and $e$ is the extension or compression (in metres, $\text{m}$ ). This equation assumes the spring has not been stretched beyond the limit of proportionality.

Just as with kinetic energy, two features need care:

the $\dfrac{1}{2}$ at the front — you must halve;
the $e^2$ — the extension is squared, so doubling the extension quadruples the stored energy. Work out $e^2$ first, then multiply by $k$ , then halve.

The equation rearranges to make $k$ or $e$ the subject:

$E_e = \dfrac{1}{2}ke^2 \qquad k = \dfrac{2E_e}{e^2} \qquad e = \sqrt{\dfrac{2E_e}{k}}$

Worked Example 1 — Energy stored in a spring

A spring has a spring constant of $200 \text{ N/m}$ and is stretched by $0.1 \text{ m}$ . Calculate the elastic potential energy stored.

Step 1 — write the equation: $E_e = \dfrac{1}{2}ke^2$ .

Step 2 — square the extension first: $e^2 = 0.1^2 = 0.01$ .

Step 3 — substitute: $E_e = \dfrac{1}{2} \times 200 \times 0.01$ .

Step 4 — calculate: $E_e = 0.5 \times 200 \times 0.01 = 1 \text{ J}$ .

Answer: the spring stores $1 \text{ J}$ of elastic potential energy.

Worked Example 2 — A stiffer spring

A spring of spring constant $800 \text{ N/m}$ is compressed by $0.05 \text{ m}$ . Calculate the elastic potential energy stored.

Step 1 — write the equation: $E_e = \dfrac{1}{2}ke^2$ .

Step 2 — square the compression: $e^2 = 0.05^2 = 0.0025$ .

Step 3 — substitute: $E_e = \dfrac{1}{2} \times 800 \times 0.0025$ .

Step 4 — calculate: $E_e = 0.5 \times 800 \times 0.0025 = 1 \text{ J}$ .

Answer: the spring stores $1 \text{ J}$ of elastic potential energy.

Worked Example 3 — Rearranging for the spring constant

A spring stretched by $0.2 \text{ m}$ stores $4 \text{ J}$ of elastic potential energy. Calculate its spring constant.

Step 1 — rearrange for $k$ : $k = \dfrac{2E_e}{e^2}$ .

Step 2 — square the extension: $e^2 = 0.2^2 = 0.04$ .

Step 3 — substitute: $k = \dfrac{2 \times 4}{0.04} = \dfrac{8}{0.04}$ .

Step 4 — calculate: $k = 200 \text{ N/m}$ .

Answer: the spring constant is $200 \text{ N/m}$ .

Worked Example 4 — Rearranging for the extension

A spring of spring constant $500 \text{ N/m}$ stores $2.5 \text{ J}$ of energy. Calculate its extension.

Step 1 — rearrange for $e$ : $e = \sqrt{\dfrac{2E_e}{k}}$ .

Step 2 — substitute: $e = \sqrt{\dfrac{2 \times 2.5}{500}} = \sqrt{\dfrac{5}{500}}$ .

Step 3 — simplify inside the root: $\dfrac{5}{500} = 0.01$ .

Step 4 — take the square root: $e = \sqrt{0.01} = 0.1 \text{ m}$ .

Answer: the spring is extended by $0.1 \text{ m}$ .

Exam Tip: Keep the extension in metres (so $10 \text{ cm}$ becomes $0.1 \text{ m}$ ) and square it first. As with kinetic energy, finding the extension from the energy needs a final square root: $e = \sqrt{\dfrac{2E_e}{k}}$ .

The Effect of Doubling the Extension

Because the extension is squared, stretching a spring twice as far stores four times as much energy, not twice as much. The table below shows the energy stored in a spring of spring constant $200 \text{ N/m}$ at different extensions.

Extension / $\text{m}$	$e^2$	$E_e = \dfrac{1}{2}ke^2$ / $\text{J}$
0.1	0.01	1.0
0.2	0.04	4.0
0.3	0.09	9.0
0.4	0.16	16.0

Elastic Potential Energy

Elastic Potential Energy

What Is Elastic Potential Energy?

Linking to Hooke's Law

The Elastic Potential Energy Equation

Worked Example 1 — Energy stored in a spring

Worked Example 2 — A stiffer spring

Worked Example 3 — Rearranging for the spring constant

Worked Example 4 — Rearranging for the extension

The Effect of Doubling the Extension

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