Conservative Forces & Potential Energy

Kinetic energy that disappears and later comes back must have been stored somewhere. Potential energy is the name for that store, and it only exists for certain kinds of force.

Work inside a system#

Two oppositely charged beads on a smooth table attract each other. No outside force acts, yet both speed up and the system’s kinetic energy grows. How? Each bead does work on the other. The two forces are equal and opposite, but so are the displacements (each moves toward the other), so both works are positive and they add up.

Energy that comes back#

Now let the beads attract each other but start out flying apart. They slow down (K falls), stop, turn round, and come back together. When they return to their starting separation, they have exactly their original speeds again. The kinetic energy wasn’t lost, only stored, and the amount stored depended only on the configuration (the separation). We call that configuration-dependent energy potential energy.

Conservative and non-conservative forces#

Compare two round trips:

• Drag a block a distance $l$ across a rough table and back. Friction opposes you both ways: it does $-\mu mgl$ each way, $-2\mu mgl$ in total. That energy is gone for good.
• Let a block oscillate on a spring. Without friction it passes the centre at the same speed every time; with friction, a little slower each pass until it stops.

Defining potential energy#

For a conservative internal force, define the change in potential energy as the negative of the work it does:

Three points follow:

• Only changes in U are defined. You may pick any configuration as U = 0, just as you choose where to put the origin.
• No potential energy can be defined for friction: its work depends on the path, so there is no single “stored” amount.
• In a rigid body the distances between particles never change, and the internal forces do no net work (their works cancel in pairs). So U only changes when separations change.

These two lines are all you need for the next lesson.