Gravitational Potential Energy — Definition

Definition

Imagine you lift a book from the floor to a shelf. You're doing work against gravity, and that work isn't lost; it's stored in the book as gravitational potential energy. This stored energy gives the book the 'potential' to do work if it falls back down.

In a more general sense, gravitational potential energy (GPE) is the energy an object possesses because of its position within a gravitational field. Think of it as the energy 'locked up' in the system of the object and the massive body (like Earth) creating the gravitational field.

The key idea is that gravity is a 'conservative force,' meaning the work done by gravity depends only on the initial and final positions, not on the path taken. This allows us to define a potential energy.

For objects near the Earth's surface, we often use the simpler formula $U = mgh$, where $m$ is the mass, $g$ is the acceleration due to gravity, and $h$ is the height above a chosen reference level (like the ground). Here, we typically set the ground as the zero potential energy level. However, this formula is an approximation valid only for small changes in height where $g$ can be considered constant.

For larger distances, or when considering objects far from Earth, we need a more fundamental definition. In this case, the reference point for zero potential energy is taken to be at 'infinity' – a point so far away that the gravitational force between the objects is negligible.

From this perspective, the gravitational potential energy of two masses $M$ and $m$ separated by a distance $r$ is given by $U = -\frac{GMm}{r}$. The negative sign is crucial: it indicates that the gravitational force is attractive.

As objects get closer (smaller $r$), their potential energy becomes more negative, meaning the system is more tightly bound. To separate them (increase $r$), you need to do positive work, increasing their potential energy towards zero.

At infinity, $r \rightarrow infty$, so $U \rightarrow 0$, which aligns with our chosen reference.