a conservative force

Imagine carrying a book up to a shelf and back down to where you started. Gravity took energy from you on the way up and gave exactly that much back on the way down — the round trip cost nothing. A conservative force is one that behaves this way: whatever energy it takes on one path, it returns on the reverse, so a complete loop comes out even.

Precisely, a force is conservative if the work it does moving an object between two points depends only on those two points — the start and the end — and not at all on the route taken. An equivalent test: the net work it does around any closed loop is exactly zero. This special property is what lets us invent a potential energy U for the force, defined so that the work done equals minus the change in potential energy, W = -ΔU. No such energy can be defined for a force that fails the test.

Gravity, the pull of an ideal spring, and the electric force between charges are all conservative. Friction is the classic counterexample: drag a block in a circle back to the start and friction has stolen energy the whole way, never returning any, so its loop does not come out even. This distinction matters enormously, because mechanical energy is conserved only when the forces doing work are conservative.