# 4.4 Conservative forces and potential energy

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• Define conservative force, potential energy, and mechanical energy.
• Use the work-energy theorem to show how having only conservative forces implies conservation of mechanical energy.

## Potential energy and conservative forces

Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force    is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy     $\left(\text{PE}\right)$ for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative . That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.

## Potential energy and conservative forces

Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable.

A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken.

We can define a potential energy $\left(\text{PE}\right)$ for any conservative force. The work done against a conservative force to reach a final configuration depends on the configuration, not the path followed, and is the potential energy added.

## Conservation of mechanical energy

Let us now consider what form the work-energy theorem takes when only conservative forces are involved. This will lead us to the conservation of energy principle. The work-energy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy. In equation form, this is

${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{{\text{mv}}_{0}}^{2}=\Delta \text{KE.}$

If only conservative forces act, then

${W}_{\text{net}}={W}_{\text{c}}\text{,}$

where ${W}_{c}$ is the total work done by all conservative forces. Thus,

${W}_{\text{c}}=\text{Δ}\text{KE.}$

Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, ${W}_{\text{c}}=-\text{Δ}\text{PE}$ . Therefore,

$-\text{Δ}\text{PE}=\text{Δ}\text{KE}$

or

$\text{Δ}\text{KE}+\text{Δ}\text{PE}=0.$

This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,

where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy    principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy    , $\left(\text{KE}+\text{PE}\right)$ . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between $\text{KE}$ and the various types of $\text{PE}$ , with the total energy remaining constant.

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