Elastic potential energy

Like weight, the elastic force of a spring also has an associated potential energy. When we compress or stretch a spring, the work we spend deforming it does not vanish: it is stored in the spring as elastic potential energy, ready to be given back as the spring returns towards its rest length.

Principle — Elastic potential energy

$\ev{E_{\text{pot,molla}} = \frac{1}{2}\,K\,(\ell - \ell_0)^2}$ where $K$ is the spring constant and $\ell_0$ the rest length of the spring.

The intuition behind the formula is illuminating. What matters is not the length $\ell$ of the spring by itself, but its deformation $\ell - \ell_0$, i.e. how far it departs from the rest length $\ell_0$. At rest ($\ell = \ell_0$) the deformation is zero and the elastic energy is zero: the spring has nothing to give back. As we compress or stretch it, the energy grows.

Two details deserve attention. First, the deformation appears squared: compressing or stretching by the same amount stores the same energy (the sign of $\ell - \ell_0$ is irrelevant), and doubling the deformation quadruples the energy. Second, the spring constant $K$ measures the “stiffness” of the spring: for the same deformation, a stiffer spring (large K) stores more energy. It is the same $K$ that appears in Hooke’s law for the force, and this is no coincidence: the elastic energy is precisely the work accumulated deforming the spring against that force.

Links

Topics: Lavoro ed energia Concepts: Energia potenziale elastica · Forza elastica e legge di Hooke Objects: Molla

Related exercises: Worked exercise — apple and spring · Problem — Spring constant from a graph · Problem — Ball launched by a spring