Summary

General Principles

Two variations of the energy principle are

\begin{array}{l} Δ E_{sys} = Δ K + Δ U + Δ E_{th} = W_{ext} \\ K_{i} + U_{i} + W_{ext} = K_{f} + U_{f} + Δ E_{th} \end{array}

Model Define the system.

Visualize Draw a before-and-after pictorial representation and an energy bar chart.

Solve Use the energy principle:

K_{i} + U_{i} + W_{ext} = K_{f} + U_{f} + Δ E_{th}

Review Is the result reasonable?

Isolated system: $W_{ext} = 0$ . The total system energy $E_{sys} = K + U + E_{th}$ is conserved. $Δ E_{sys} = 0$ .
Isolated, nondissipative system: $W_{ext} = 0 and Δ E_{th} = 0$ . The mechanical energy $E_{mech} = K + U$ is conserved: $K_{i} + U_{i} = K_{f} + U_{f}$ .

Potential energy, or interaction energy, is energy stored inside a system via interaction forces. The energy is stored in fields.

Potential energy is associated only with conservative forces for which the work done is independent of the path.
Work $W_{int}$ by the interaction forces causes $Δ U = - W_{int}$ .
Force $F_{s} = - d U / d s = - (slope of the potential energy curve)$ .
Potential energy is an energy of the system, not an energy of a specific object.

Energy diagrams show the potential-energy curve PE and the total mechanical energy line TE.

From the axis to the curve is $U$ . From the curve to the TE line is K.
Turning points occur where the TE line crosses the PE curve.
Minima and maxima in the PE curve are, respectively, positions of stable and unstable equilibrium.

Gravitational potential energy is an energy of the earth $+$ object system:

U_{G} = m g y

Elastic potential energy is an energy of the spring $+$ attached objects system:

U_{Sp} = \frac{1}{2} k {(Δ s)}^{2}

Energy bar charts show the energy principle in graphical form.