Conservative Forces and Potential Energy

Potential Energy Meaning

Before we dive deep into the specifics of potential energy, let's start by talking about energy in general.

Recall that work is equal to force times the displacement. Written as an integral, it looks like this:

$$W=\int_{x_\mathrm{i}}^{x_\mathrm{f}} F(x) dx\mathrm{.}$$

Here, we see that the work can be calculated by doing the force \(F(x)\) times the displacement in increments of \(\mathrm{d} x\) over the total distance we need to cover (from \(x_\mathrm{i}\) to \(x_\mathrm{f}\)). This means that an object, or a system, has more energy, the more it can exert a force over a certain displacement.

We can describe this theorem mathematically as:

$$W=\Delta KE$$

where \(W\) is the work done and \(\Delta KE\) is the change in the kinetic energy. Remember that the kinetic energy is equal to

$$KE = \frac{1}{2}\\mv^2\mathrm{,}$$

therefore, we can rewrite our work equation by substituting the above equation for our kinetic energy. Our new equation would be

$$W=\frac{1}{2}\\mv_\mathrm{f}^2-\frac{1}{2}mv_\mathrm{i}^2\mathrm{.}$$

This is important because it will help us to understand the relationship between potential energy, kinetic energy, and work.

Potential Energy Definition

For us to really move forward in the Potential Energy and Conservative Forces article, we should probably define potential energy.

For example, a pencil sitting on the ground would have no potential energy. However, if I were to pick that pencil up and raise it a meter off the ground, it would have potential energy. But why? What changed?

Now that the pencil is suspended in the air, the force of gravity wants to act on it, and the second I let go, that pencil will come crashing down. Therefore, when I raise the pencil above the floor, I give it the potential to do work. Before, it could do nothing; it was just sitting there. But now, it has the ability, if released, to unleash work by transferring potential energy into kinetic energy. Remember the work-energy theorem from above? As the pencil falls to the ground, it will have a velocity, which gives it kinetic energy, which due to the work-energy theorem, allows it to do work.

Fig. 2 - At the top of the ramp, all the energy that the ball has is potential. Once it gets to the bottom of the ramp, all the ball's energy has been converted from potential to kinetic energy.

Potential Energy Formula

Ok, so I know what potential energy is, but how do I quantify it? That is what the potential energy formula is for. With the formula

$$\Delta U_\mathrm{g} = mg\Delta h\mathrm{,}$$

we can determine how much gravitational potential energy an object has.

Notice a few things.

• Potential energy is represented by a capital \(U\).
• The little \(\mathrm{g}\) subscript means that this formula is for gravitational potential energy only.
• \(m\) is the mass, \(g\) is the gravitational constant, and \(\Delta h\) is the change in height (so in the pencil example, it would be \(1\,\mathrm{meter}\) because the pencil was \(1\,\mathrm{meter}\) off the ground).

Conservative Force and Potential Energy

We couldn't talk about potential energy without talking about conservative forces. Don't worry; physics cares nothing for politics. We aren't about to get into a rant about the second amendment or anything. We will, however, keep the theme a little patriotic and talk about independence, well, path independence at least. (Yeah, I know your eyes probably aren't going to recover from rolling so much because of those terrible puns).

What is a Conservative Force?

In a nutshell, conservative forces are forces that are independent of path. This means that we don't care about the middle of the story. If I was a strict boss and wanted you to work at 9:00 a.m. sharp, I wouldn't care how you got there, just that you did get there. I don't care if you took the bus, car, train, or plane: as long as you're there at 9:00 a.m. That's how conservative forces work.

Fig. 3 - Notice how the path taken does not matter; either way, we're getting from \(E=0\) to \(E=mgh\).

It is also essential to make note of another type of force: dissipative forces.

Therefore, they are different than conservative forces in that respect. Examples of conservative forces are friction, resistive forces, or forces outside the system.

Relation Between Conservative Force and Potential Energy

Potential Energy allows for work to be done independent of path. Therefore, you can think of it like this: potential energy "sets up" the perfect scenario for conservative forces, giving them something of an alley-oop.

By understanding the relationship between conservative forces and potential energy, we can derive a formula for an object-earth system:

$$U_\mathrm{g} = \frac{-Gm_1 m_2}{r}\\\mathrm{.}$$

Notice here the similarities between this equation and the potential energy equation above.

Application of Conservative Force and Potential Energy

By understanding the relationship between conservative forces and potential energy, mathematicians and physicists devised this fancy mathematical relationship:

$$\Delta U = -\int_a^b \vec F_{\mathrm{cf}} \cdot \mathrm{d} \vec r\mathrm{.}$$

We could really go down the rabbit hole and go into the derivation of this formula, but I'll spare you the mathematical mess. This means that potential energy is equal to the integral of the conservative force times the displacement. Does that sound familiar? It sounds similar to our formula for work! It's almost like work and potential energy are related...wait, isn't energy just the capacity to do work?

If you were to differentiate that integral to figure out the force with respect to the potential energy, the formula

$$F_x = \frac{-\mathrm{d}U(x)}{\mathrm{d}x}\\$$

would result.

Conservative Force and Potential Energy Examples

An ideal spring is an example of a conservative force, whose force is represented by the equation

$$\vec F_\mathrm{s} = -k\Delta \vec x\mathrm{.}$$

The force with the underscore \(s\) means the force of the spring, \(k\) is the spring constant, and \(x\) is the displacement of the spring.

The formula

$$U_\mathrm{s} = \frac{1}{2} k \Delta x^2$$

is used to calculate the potential energy in a spring.

We've come a long way since the first of this article. You've just been overloaded with information; now it's clutch-time. We are on the green and it's time to put the ball in the hole. These are the most essential concepts you will need to remember.