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 ($m_1$ and $m_2$) with respect to distance. Given the force:

${𝐅}_g=\frac{G{m}_1{m}_2}{r^2}$

We integrate to get potential energy:

$U_g=\int {𝐅}_{g}dr=\int \frac{G{m}_1{m}_2}{r^2}dr=-\frac{G{m}_1{m}_2}{r}$

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 $m_2$ is the mass of the planet):

${𝐅}_g=\frac{G{m}_1{m}_2}{r^2}$

$g=\frac{G{m}_2}{r^2}$

and then puting it into $U_g$:

$U_g=-m_{1}gr$

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

$U_g=mgr$

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 ($k$) and the displacement from the rest position ($x$). It should be noted that the amount of energy is the same regardless whether the object is compressed or elongated. Given the force:

${𝐅}_s=-k𝐱$

We integrate to get energy:

$U_s=\int {𝐅}_{s}dx=\int -k𝐱dx=-\frac{1}{2}k{𝐱}^2$