Definition: Potential Energy

Roughly speaking, a conservative force is a force that conserves energy, i.e. it does not result in dissipation of energy to heat or other forms. For a more precise definition, a conservative force is such that the work done by that force is independent of an object's path. Gravity and spring forces are conservative forces, while friction is not. For conservative forces, the potential energy associated with a force is equal to negative the work done by the force over some distance. In other words, $$\Delta P=-W$$ For gravity, potential energy is $P=mgh$ (where $h$ is the height with respect to a reference point), and for springs the potential energy is $P=(1/2)kx^2$ (where $x$ is the stretch/compression of the spring).

Theorem: Conservation of Energy

The total change in energy, $$\Delta U =\Delta KE + \Delta PE,$$ is equal to 0 (conserved) if there are no external non-conservative forces. If there are non-conservative forces, teh statement is revised to $$\Delta U = \Delta KE + \Delta PE = W_{\text{nc}}.$$

Idea: Potential Energy Diagrams

Potential energy diagrams can give insight into the motion of an object — equilibria, turning points, max velocities, etc. Here are some of the key features:

• Local minima — stable equilibria
• Local maxima — unstable equilibria
• When $E = U(x)$ — turning points
• Negative of slope — force

It's nice to think of the whole diagram like a hill -- the object can rock between the turning points.