Potential energy

Conservative force

$$\begin{align*} \Delta U &= - \int_a^b\vec F\cdot d\vec s \\ F &= -\frac{dU}{dx} \end{align*}$$

Gravitational PE

Close to Earth's surface:

$$U_{grav} = mgh$$

Away from Earth's surface:

$$U_{grav} = -\frac{GM_Em}{r}$$

Elastic PE (Hooke's law)

$$\begin{align*} F_s &= -kx \\ U_e &= \frac{1}{2}kx^2 \end{align*}$$

• $F_s$: spring restoring force (negative: opposite to the displacement)
• $k$: spring constant

PE-position graph

• $x_1, x_3$ are points of minima: stable equilibrium
• $x_2, x_4$ are points of maxima: unstable equilibrium