Rolling Without Slipping

Picture a wheel rolling along flat ground with no skidding. At every instant, one point on the rim is touching the ground, and that point has zero velocity. That feels strange since the wheel is clearly moving, but the motion of any point on the wheel is the sum of two pieces: the center moving forward at speed v, and the point spinning around the center at speed omega r. At the bottom, the spin velocity points backward and exactly cancels the forward motion of the center, leaving zero. This is the no-slip condition, and it forces v = omega r: the faster the center moves, the faster the wheel must spin to keep that contact point still.

Follow one point on the rim over a full turn and it traces a looping curve called a cycloid. At the bottom of each loop the point momentarily stops (that's the contact instant), then it speeds up, reaching its fastest at the top of the wheel where its spin velocity and the center's forward velocity point the same way and add up to 2v. The point never moves backward; it just slows nearly to a stop, loops up and over, and comes back down.

This matters for energy because the static friction force at that contact point does zero work. Work needs motion of the point the force acts on, and that point isn't moving. That's why rolling without slipping lets you use energy conservation freely, with no friction losses to track, even though friction is what keeps the wheel from sliding and is what changes its spin rate when something speeds it up or slows it down. This sets up rolling races down ramps, where rotational and translational kinetic energy trade off cleanly.