View source for Physics: Calculus-based:Conservation of energy:Potential energy

Potential energy is the energy stored in an object due to its position. There are several types of potential energy.

== Gravitational ==

Gravitational potential energy, as all energy functions, is the integral of force between two objects (<math>m_1</math> and <math>m_2</math>) with respect to distance. Given the force:

<math>\mathbf{F}_g = \frac{Gm_1 m_2}{r^2}</math>

We integrate to get potential energy:

<math>U_g = \int \mathbf{F}_g\,dr = \int \frac{Gm_1 m_2}{r^2}\,dr = -\frac{Gm_1 m_2}{r}</math>

Sometimes, when dealing with small distances where the difference in acceleration due to gravity will be negligable we simplify the energy equation by rearranging the force equation (where <math>m_2</math> is the mass of the planet):

<math>\mathbf{F}_g = \frac{Gm_1 m_2}{r^2}</math>

<math>g = \frac{Gm_2}{r^2}</math>

and then puting it into <math>U_g</math>:

<math>U_g = -m_1 gr</math>

In many cases, this is further simplified by removing the negative sign, yeilding:

<math>U_g = mgr</math>

== Elastic ==

Elastic potential energy is the energy stored in a compressed or elongated object (a spring, for example). The amount of energy stored in the object depends on spring constant (<math>k</math>) and the displacement from the rest position (<math>x</math>). It should be noted that the amount of energy is the same regardless whether the object is compressed or elongated. Given the force:

<math>\mathbf{F}_s = -k\mathbf{x}</math>

We integrate to get energy:

<math>U_s = \int \mathbf{F}_s\,dx = \int -k\mathbf{x}\,dx = -\frac{1}{2}k\mathbf{x}^2</math>

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