7.3 Gravitational potential energy  (Page 2/5)

Converting between potential energy and kinetic energy

Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to $\text{mgh}$ on it, thereby increasing its kinetic energy by that same amount (by the work-energy theorem). We will find it more useful to consider just the conversion of ${\text{PE}}_{\text{g}}$ to $\text{KE}$ without explicitly considering the intermediate step of work. (See [link] .) This shortcut makes it is easier to solve problems using energy (if possible) rather than explicitly using forces.

More precisely, we define the change in gravitational potential energy $\text{Δ}{\text{PE}}_{\text{g}}$ to be

where, for simplicity, we denote the change in height by $h$ rather than the usual $\text{Δ}h$ . Note that $h$ is positive when the final height is greater than the initial height, and vice versa. For example, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is

Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly considering the force of gravity that does the work .

Using potential energy to simplify calculations

The equation $\text{Δ}{\text{PE}}_{\text{g}}=\text{mgh}$ applies for any path that has a change in height of $h$ , not just when the mass is lifted straight up. (See [link] .) It is much easier to calculate $\text{mgh}$ (a simple multiplication) than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position $h$ of a mass $m$ is accompanied by a change in gravitational potential energy $\text{mgh}$ , and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force.