Force and Potential Energy

Relation between Forces and Potential Energy

For a system to have potential energy, there must be one or more conservative forces acting on objects in the system. A conservative force is a force by which the work done is path-independent and reversible.

When a conservative force like gravity works on an object, potential energy is stored that can be converted to kinetic energy to later reverse the work done. When a non-conservative force such as friction works on an object, kinetic energy converts to thermal energy, and we can not get the dissipated thermal energy back. Thus, potential energy is only stored in the system when there is a conservative force acting on objects in the system. If only non-conservative forces are acting on objects in the system, there is no potential energy in the system.

Types of Forces that Provide Potential Energy

A few common conservative forces we use in physics problems are the force of gravity, the spring force, and the electric force. As mentioned above, when these forces act on objects in a system, the system has potential energy. Some examples of non-conservative forces are friction, air resistance, and the pushing/pulling force. When these forces act on objects in a system, potential energy is not stored, but rather some energy is lost to other forms of energy like heat. More detail regarding conservative and non-conservative forces is given in the articles, "Conservative Forces" and "Dissipative Forces".

Formula for Forces and Potential Energy

In order to develop a formula that relates the conservative forces acting on objects in a system with the potential energy, let’s consider how the work done by the forces relates to the potential energy. When work is done on objects in a system, the objects experience a displacement. If the work was done by a conservative force, there will be a change in potential energy from the initial location compared to the final location of the object. The work done by conservative forces, \(W\), is equal to minus the change in potential energy, \(-\Delta U\), of the system:

\[W=-\Delta U.\]

We recall that the work done by a force is found by multiplying the force by the displacement if it is a constant force, and if it is a varying force we take the integral of the force with respect to distance:

\[W=\int_\vec{a}^\vec{b}\vec{F}(\vec{r})\cdot\mathrm{d}\vec{r}.\]

In this equation, \(\vec{F}(\vec{r})\) is the force vector, \(\vec{r}\) is the distance vector, and \(\vec{a}\) and \(\vec{b}\) are the initial and final positions. To simplify the problem a bit, we will just consider motion in one spatial dimension, so we will use:

\[W=\int_{x_1}^{x_2}F(x)\,\mathrm{d}x.\]

Substituting this into our equation above gives us:

\[\Delta U=-\int_{x_1}^{x_2}F(x)\,\mathrm{d}x.\]

We use this equation when we are trying to solve for the change in potential energy of a system. Sometimes we are given the function for the potential energy instead, and in that case we would want to solve for the force function. In this case, we will approximate that \(W=F(x)\Delta x,\) where \(\Delta x\) is a small change in distance. This is an approximation because \(F(x)\) could vary a bit over the change in distance. Now we will substitute that into our first equation relating work and the change in potential energy: \[\begin{align*}W&=-\Delta U\\F(x)\Delta x&=-\Delta U\\F(x)&=\frac{-\Delta U}{\Delta x}.\end{align*}\]

If we consider a very small change in distance, we can take the limit as \(\Delta x \to 0.\) Then our equation becomes: \[\begin{align*}F(x)&=\lim_{\Delta x \to 0}\Big(-\frac{\Delta U}{\Delta x}\Big)\\&=-\frac{\mathrm{d}U(x)}{\mathrm{d}x}.\end{align*}\]This is no longer an approximation because there is no variation in \(F(x)\) in the limit that \(\Delta x \to 0.\) We see from this relation that the force of an object can be found by taking minus the slope of the function for potential energy with respect to position.

Forces and Potential Energy Functions

In the previous section, we found functions for the change in potential energy of a system and the conservative force acting on an object in the system. To summarize, these functions are:

\[\begin{align} \Delta U&=-\int_{x_1}^{x_2}F(x)\,\mathrm{d}x,\\ F(x)&=-\frac{\mathrm{d} U(x)}{\mathrm{d} x}.\end{align}\]

Unit Analysis of Potential Energy

In one of the examples above, we found the gravitational potential energy to be given by \(U=mgh\). Let’s consider the units of that quantity to determine the units for potential energy. The mass of an object is represented by \(m\), and its SI unit is\(\mathrm{kg}\). The gravitational acceleration is represented by \(g\), and its SI unit is \(\frac{\mathrm{m}}{\mathrm{s}^2}\). Also, because \(h\) is the displacement, its SI unit is \(\mathrm{m}\). From \(U=mgh\), we see that the units of gravitational potential energy are

\[\mathrm{kg}\frac{\mathrm{m}}{\mathrm{s}^2}\mathrm{m}=\mathrm{J}.\]

Thus, the SI unit of potential energy is the joule, \(\mathrm{J}\). All energy, like work and kinetic energy, has units of joules, so potential energy does as well.

Potential Energy and Forces Graph

It is useful to graph the potential energy as a function of position. As mentioned above, minus the slope of the potential energy function, or the tangent line to the function at a certain point, gives us the force at that point. We can also discover physical properties of the system by looking at this graph, such as whether the system is in stable equilibrium. More detail is given on this in the article, "Potential Energy and Graphs".

Minus the slope of the potential energy of a spring as a function of position gives us the force, StudySmarter Originals

The graph above shows the potential energy and force of a spring as functions of position. Notice how at each position, the value of the force is minus the slope of the line tangent to the potential energy curve.