Potential Energy, and the Potential Energy of a Spring

Potential energy, $U$, is energy a body possesses because of its position or its state of configuration, rather than because of its motion, and it can be defined only for a conservative force. For a conservative force $\vec F$, the potential energy is defined so that the work done by that force as a body moves from a chosen reference configuration to its present one equals exactly the decrease in potential energy:

The minus sign is deliberate and important: when a conservative force does positive work on a body (helping it move in the direction the force already pushes), the body's potential energy decreases -- gravity doing positive work on a falling body corresponds to a decrease in its gravitational potential energy, energy that reappears as an increase in kinetic energy.

Gravitational potential energy near the Earth's surface. Taking the ground (or any convenient reference level) as the zero of potential energy, the gravitational potential energy of a body of mass $m$ at height $h$ above that reference level is $U = mgh$, obtained directly from the work gravity would do, $mgh$, in lowering the body from height $h$ down to the reference level.

Elastic potential energy of a spring. For an ideal spring obeying Hooke's law, with force constant $k$, an external agent does work $\tfrac12 kx^2$ in slowly stretching (or compressing) the spring by a distance $x$ from its natural length (derived in §5.3 by integrating the spring's restoring force $F(x) = kx$). Since the spring's own restoring force is conservative, this work done against the spring's restoring force is stored, and fully recoverable, as elastic potential energy:

$U(x) = \frac{1}{2}kx^2$ …