# Physics with Calculus/Mechanics/Energy and 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, involves the line integral of the force between two objects ($m_{1}$ and $m_{2}$ ). By Newton's universal law of gravity, the force is

$\mathbf {F} _{g}=-{\frac {Gm_{1}m_{2}}{r^{2}}}\;{\hat {r}}$ We integrate to get potential energy:

$U_{g}(r)=-\int _{\infty }^{r}\mathbf {F} _{g}\,dr'=\int _{\infty }^{r}{\frac {Gm_{1}m_{2}}{r^{2}}}\,dr=-{\frac {Gm_{1}m_{2}}{r}}$ Here, we have taken the reference point (where the potential energy equals zero) to be at $r=\infty$ . Sometimes, when dealing with small distances where the difference in acceleration due to gravity will be negligeable we simplify the energy equation by assuming that $r=R+y$ , where $R$ is the Earth's radius and $y< is the height above the Earth's surface. Taking $m_{2}$ to be the mass of the planet:

$F_{g}={\frac {Gm_{1}m_{2}}{R^{2}}}$ $g={\frac {Gm_{2}}{R^{2}}}$ .

Note that the vector $g$ points in the $-{\hat {r}}$ direction. Inserting this into the integral for $U_{g}$ :

$U_{g}=-\int _{0}^{y}(-m_{1}g{\hat {r}})dr'=m_{1}gy$ ,

where now, the reference point is on the surface of the Earth.

## 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:

$\mathbf {F} _{s}=-k\mathbf {x}$ We integrate to get energy:

$U_{s}=-\int \mathbf {F} _{s}\,dx=\int -k\mathbf {x} \,dx={\frac {1}{2}}k\mathbf {x} ^{2}$ 