Potential energy for a linear spring

A horizontal spring exerts a force F = (−kx, 0, 0) that is proportional to its deformation in the axial or x direction. The work of this spring on a body moving along the space curve s(t) = (x(t), y(t), z(t)), is calculated using its velocity, v = (v_x, v_y, v_z), to obtain

${\displaystyle W={\int <!--\; \int \; -->}_0^t\mathbf{F}\cdot <!--\; \cdot \; -->\mathbf{v}dt=-<!--\; -\; -->{\int <!--\; \int \; -->}_0^{t}kx{v}_{x}dt=-<!--\; -\; -->{\int <!--\; \int \; -->}_0^{t}kx\frac{\mathrm{d}x}{\mathrm{d}t}dt={\int <!--\; \int \; -->}_{x(t_0)}^{x(t)}kx\mathrm{d}x=\frac{1}{2}k{x}^2}$

For convenience, consider contact with the spring occurs at t = 0, then the integral of the product of the distance x and the x-velocity, xv_x, is x2/2.

The function

${\displaystyle U(x)=\frac{1}{2}k{x}^2,}$

is called the potential energy of a linear spring.

Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy.