Spring elastic energy:
formula and worked example

What stored spring energy represents

When you compress or stretch a spring, the work you do is stored as elastic potential energy. Release it and that energy comes back as motion, which is how a toy launcher, a pen click, a door closer or a vehicle suspension turns stored energy into action.

The number answers a practical question: how much energy is available to throw a projectile, absorb an impact or hold in reserve? Knowing the joules lets you compare a spring with other energy sources such as falling weights or batteries.

Formula and the meaning of each input

• Spring constant: the stiffness in newtons per metre. A higher value means a stiffer spring.
• Extension: the distance stretched or compressed from the natural length, in metres. It is not the total length of the spring.

The half in the formula appears because force rises steadily from zero as the spring is stretched, so the average force during the stretch is half of the final force. Energy equals average force times distance.

The formula holds as long as the spring obeys Hooke's law, meaning the force is proportional to extension. Past the elastic limit the spring deforms permanently and the relationship no longer applies.

Stretching a 500 N/m spring by 8 cm

1. 1Convert the extension: 8 cm = 0.08 m.
2. 2Square it: 0.08 × 0.08 = 0.0064 m².
3. 3Multiply by the spring constant: 500 × 0.0064 = 3.2.
4. 4Take half: 0.5 × 3.2 = 1.6 J.

Check by force: at full stretch the force is 500 × 0.08 = 40 N. Average force is 20 N over 0.08 m, giving 20 × 0.08 = 1.6 J, in agreement.

What 1.6 joules means in practice

The value is small. It is roughly the energy needed to lift a 1 kg object by 16 cm. For many mechanisms that is sufficient: a small spring-loaded latch uses a fraction of a joule.

• Release into motion: with no losses, 1.6 J would give a 0.2 kg mass a speed of about 4 m/s.
• Energy in a compressed spring is the same as when stretched by the same amount.
• Real springs lose a little to heat, internal friction and sound, so delivered energy is less.

Treat the figure as the maximum available. Whatever is attached, friction and the spring's own mass reduce what you actually receive.

How stiffness and extension push the number

A spring that is 10% stiffer, 550 N/m rather than 500, stores 1.76 J at the same extension, exactly 10% more. Stiffness acts linearly.

Extension is the far stronger lever. Raise the stretch by 10% to 0.088 m and the energy rises by 21%, since 1.1 squared is 1.21. Doubling the stretch gives 6.4 J, four times as much, while the force needed only doubles.

Typical errors with spring energy

• Using force times extension without the one half, which gives 3.2 J here.
• Forgetting to square the extension, or squaring after multiplying.
• Entering extension in centimetres into a formula calibrated for metres, which makes the answer 10,000 times too big.
• Measuring from the stretched length instead of the natural length.

Choosing and checking a spring

To size a spring for an energy budget, rearrange for extension: x = √(2E ÷ k). For 1.6 J and 500 N/m this gives 0.08 m. Then confirm the spring can reach that deflection without exceeding its safe working range.

Remember that two springs side by side add their constants, while two in a chain are softer than either alone. Compute the combined constant first, then the energy.

Springs are compact stores only for small amounts of energy. Next to a battery, 1.6 J is minute, so springs suit short, repeated bursts such as latches, dampers and launchers rather than long-duration supply. Their strength is that the energy returns almost instantly and can be released many thousands of times.