Gravitational Potential Energy

What gravitational potential energy is

A body held at height $h$ above the ground has the capacity to do work as it falls — it can drive a turbine, compress a spring, or accelerate another body. This stored capacity is called gravitational potential energy:

$U = mgh$

It is energy by virtue of position, not motion.

Derivation

Potential energy is defined through the work done by the conservative force:

$\Delta U = -W_{gravity}$

Work done by gravity when body falls through height $h$: $W_{gravity} = mgh$

Therefore: $\Delta U = -mgh$

Taking $U = 0$ at $h = 0$ (ground level):

$U(h) - U(0) = -mgh \implies U(h) = mgh$

$\boxed{U = mgh}$

The reference level is arbitrary

The formula gives the potential energy relative to the chosen reference level ($h = 0$). The choice of reference is arbitrary — only changes in PE are physically meaningful.

• If ground is reference: $U = mgh$
• If table (height $H$) is reference: $U = mg(h - H)$

The change in PE between two points is the same regardless of reference:

$\Delta U = mg(h_2 - h_1) = mgh_2 - mgh_1$

PE and work done against gravity

To lift a body from $h_1$ to $h_2$ at constant velocity (no change in KE), the external agent must do work against gravity:

$W_{external} = mg(h_2 - h_1) = \Delta U$

This work is stored as gravitational PE. The agent has transferred energy to the gravitational field.

Conversion between PE and KE

When a body falls freely from height $h$:

$U_{top} \to KE_{bottom}$

$mgh = \frac{1}{2}mv^2 \implies v = \sqrt{2gh}$

This is conservation of mechanical energy — PE converts to KE as the body falls.

Limitations

$U = mgh$ assumes $g$ is constant. This holds near Earth's surface (within a few kilometres). For large heights, the correct formula is:

$U = -\frac{GMm}{r}$

where $r$ is the distance from Earth's centre. The negative sign and the $1/r$ dependence replace the linear $mgh$ formula for large distances.