Potential Force Field. Potential Energy

A potential force field is one in which the work done between two positions is independent of the path and depends only on the initial and final positions. Such forces are called conservative.

Potential energy

For a conservative force, potential energy $\Pi$ is defined so that:

$$A_{1\to2}=\Pi_1-\Pi_2=-\Delta\Pi.$$

The zero level of potential energy is arbitrary; only differences in potential energy have physical significance.

Force and potential energy

In three dimensions, a conservative force is related to potential energy by $\vec F=-\nabla\Pi$. In one-dimensional motion this becomes $F_x=-d\Pi/dx$.

Gravity near Earth's surface

With height $h$ measured upward, gravitational potential energy may be written $\Pi_g=mgh+C$. Choosing zero potential at $h=0$ gives $\Pi_g=mgh$.

Elastic force

For a linear spring with $F_x=-kx$, the elastic potential energy is:

$$\Pi_s=\frac{kx^2}{2}+C.$$

It is common to choose $\Pi_s=0$ at $x=0$.

Properties of conservative forces

The work of a conservative force around any closed path is zero. Dry sliding friction and most resistance models are nonconservative because their work depends on the path traveled.

Example

A 3 kg body descends by 2 m. Its change in gravitational potential energy is $\Delta\Pi=-3g\cdot2$, while the work of gravity is $A_g=6g\approx58.9$ J for $g=9.81$ m/s².

Common mistakes

• confusing work of a conservative force with change in potential energy; their signs are opposite;
• treating the absolute value of potential energy as unique without choosing a reference level;
• assigning potential energy to dry friction;
• omitting the minus sign in $\vec F=-\nabla\Pi$.