Conservative vs. Non-Conservative Forces

Forces are classified into two categories based on how they do work:

Gravity and spring forces are the primary examples.

Consider lifting a book of mass $m$ to height $h$. Gravity does work $W_g = -mgh$ whether you lift it straight up or along any other path. Only the vertical displacement matters.

Friction and air resistance are the main examples.

Consider pushing a box across a floor with kinetic friction force $f_k = \mu_k mg$. The work done by friction is:

where $d$ is the total distance traveled. A straight path of length $d_1$ gives work $W_1 = -\mu_k mg d_1$. A longer path of length $d_2 > d_1$ between the same endpoints gives more negative work: $W_2 = -\mu_k mg d_2$. The endpoints are identical, but the work differs because the path lengths differ.