Gravitational Potential Energy

When you lift an object of mass $m$ through a vertical height $h$ in Earth's gravitational field, you do work against the gravitational force. This work is stored as gravitational potential energy in the object-Earth system.

Derivation from Work:

Consider lifting an object of mass $m$ slowly at constant velocity from a reference level (where we define $U_g = 0$, more on this later) to height $h$ above that level.

The gravitational force on the object has magnitude:

This force acts downward. To lift the object at constant velocity, you must apply an upward force equal in magnitude to the gravitational force. The displacement is upward through distance $h$.

Since the applied force and displacement are in the same direction, the work you do is:

This work is stored as gravitational potential energy in the system:

Here $g \approx 9.8~\text{m/s}^2$ near Earth's surface, and $h$ is measured from the chosen reference level.

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