Potential Energy

A work in progress by Matthew Lewine (mlewine3)

The Main Idea

Potential energy is stored energy which results from position or configuration. It is often contrasted with kinetic energy.

In terms of potential energy, its capacity for doing work is a result of its position in a gravitational field (gravitational potential energy), an electric field (electric potential energy), or a magnetic field (magnetic potential energy). It may have elastic potential energy due to a stretched spring or other elastic deformation.

The unit for energy in SI is the joule, which has the symbol J.

The universe's matter flows towards the minimum total potential energy. This cosmic flow is time.

A Mathematical Model

A force is considered conservative if it is acting on an object as a function of position only.

We can relate work to potential energy using the equation

[math]\displaystyle{ U = -\int \vec{F}\cdot\vec{dr} }[/math]

This says that the potential energy U is equal to the work you must do to move an object from an arbitrary reference point [math]\displaystyle{ U=0 }[/math] to the position [math]\displaystyle{ r }[/math]. We can take the derivative of both sides of this equation and obtain:

[math]\displaystyle{ \frac{-dU}{dx} = F(x) }[/math]

This says that the force on an object is the negative of the derivative of the potential energy function U. This means it is the negative of the slope of the potential energy curve. Plots of potential functions are valuable aids to visualizing the change of the force in a given region of space.

Let's apply this relationship. If the potential energy function U is known, the force at any point can be obtained by taking the derivative of the potential. Let's consider gravitational potential and elastic potential.

The potential energy function U of gravitational potential is [math]\displaystyle{ mgh }[/math], where [math]\displaystyle{ m }[/math] is mass, [math]\displaystyle{ g }[/math] is the gravitational constant, and [math]\displaystyle{ h }[/math] is some distance away from the reference point at which U = 0. Then the force is

[math]\displaystyle{ F = \frac{-d}{dh}mgh = -mg }[/math]

We can go the other way as well. We know the force of gravity is [math]\displaystyle{ -mg }[/math], and integrating with respect to h we obtain [math]\displaystyle{ U = mgh }[/math].

This process can be done with elastic potential as well, where the force [math]\displaystyle{ F = -kx }[/math] and the potential energy function is [math]\displaystyle{ U = \frac{1}{2}k^{2} }[/math]

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript