Result

Gravitational and spring potential energy

U_{g} = m g h, U_{s} = \frac{1}{2} k x^{2}

Statement

Energy stored by raising a mass a height h near Earth's surface, or by stretching or compressing a spring a distance x from equilibrium.

Symbols and units

$U_{g}, U_{s}$	gravitational, spring potential energy	J
$m$	mass	kg
$g$	gravitational acceleration	9.8 m/s²
$h$	height	m
$k$	spring constant	N/m
$x$	displacement from equilibrium	m

When it applies

U_g near a surface with g constant; U_s for an ideal spring obeying Hooke's law.

Don't use when

U = mgh is a near-surface approximation only, valid where g is constant over the height change — the exact gravitational form is −GMm/r.

Derivation

Potential energy is defined as the negative of the work done by a conservative force as the object moves from a reference point:

U = - \int F d x

For a constant force mg over a height h, and for a spring force −kx over a displacement x, this integrates to:

U_{g} = m g h, U_{s} = \frac{1}{2} k x^{2}

Worked examples

Spring vs height

A 0.5 kg mass raised 2.0 m:

U_{g} = m g h = (0.5) (9.8) (2.0) \approx 9.8 J

A spring of k = 200 N/m compressed 0.10 m:

U_{s} = \frac{1}{2} k x^{2} = \frac{1}{2} (200) (0.10)^{2} = 1.0 J

Watch out for

U_g = mgh is a near-surface approximation, valid only where g is constant — the exact form is −GMm/r.