Conservative Forces and Reversible Energy

The fundamental relationship $W_c = -\Delta U$ reveals that conservative forces store and release energy reversibly and path-independently. The work done depends only on initial and final positions, not on the trajectory between them.

Reversible Energy Transfer

When a conservative force does positive work, potential energy decreases by exactly that amount. If $U_f < U_i$, then:

$$W_c = -(U_f - U_i) = U_i - U_f > 0$$

Reversing the motion causes the force to do negative work of equal magnitude, restoring the original potential energy. No energy is lost.

For gravity, lowering an object from height $y_i$ to $y_f$ yields work $W_g = mg(y_i - y_f)$. Raising it back requires work $W_{\text{against}} = mg(y_f - y_i)$ of equal magnitude. The process is perfectly reversible.

Path Independence

Consider two different frictionless paths connecting points $A$ and $B$ at different heights. Along a straight ramp, gravity does work continuously. Along a wavy path with hills and valleys, gravity alternates between positive work (descending) and negative work (ascending). Yet the total work is identical:

$$W_g[\text{path 1}] = W_g[\text{path 2}] = mg(y_A - y_B) = -\Delta U_g$$

This path independence follows directly from $W_c = -\Delta U$. Since $\Delta U = U_f - U_i$ depends only on endpoints, so does the work. The intermediate details cancel out.