Potential energy

Potential energy ($U$) is stored Energy due to position or configuration. It is useful when a conservative force can give energy back as Work.

Gravitational potential energy near Earth

For small height changes near Earth’s surface:

$$U_g=mgh$$

where $h$ is height relative to your chosen zero level. The zero is arbitrary; changes in potential energy are what matter.

$$\Delta U_g=mg\Delta h$$

If an object falls, gravitational potential energy decreases and kinetic energy increases.

Spring potential energy

For an ideal spring:

$$U_s=\frac{1}{2}k{x}^2$$

where $x$ is displacement from equilibrium.

Link to work

For conservative forces:

$$W_{conservative}=-\Delta U$$

So if gravity does positive work while an object falls, gravitational potential energy decreases.

Common mistake: potential energy is not stored “in the object alone”; it belongs to the object-system setup, such as object + Earth or mass + spring.

Unit: Joule.

Choosing the zero

You are free to set $U=0$ wherever it makes the algebra clean. For a falling object, the ground is often convenient; for a spring, equilibrium is convenient. Changing the zero shifts every potential energy value by a constant, but it does not change $\Delta U$, Work, or physical predictions.