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Skill 8 of 13
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# Elastic potential energy

The elastic force $F_\text{elastic}=k\Delta L$,1 where $k$ is the spring constant, tries to make a spring (or rubber band or other elastic material) return to its relaxed length and might push other things out of the way while doing so.

Meaning, a spring has the potential to cause motion; while compressed/elongated it carries some, let’s call it elastic potential energy, symbolised with $U_\text{elastic}$.

We happen to have a formula for it (derived in Proof 37 if interested):2

$$U_\text{elastic}=\frac 12 k\,\Delta L^2$$

Note: The elastic potential energy being stored is not linear. Double the compression/elongation $\Delta L$ and you are quadrupling $U_\text{elastic}$.3 How can this be?

Because, by elongating/compressing a spring farther,

• not only can it push back over a longer distance $\Delta L$, meaning more energy is stored;
• the elastic force itself also increases!4 A larger elastic force makes things that are in the way accelerate more; it provides a larger “potential for motion”, meaning more energy stored once again.

With both of these effects happening as the spring is elongated/compressed, the stored elastic potential energy is quadratic with $\Delta L$ rather than linear.