A rolling object is doing two things at once: its center of mass is moving through space, and the object itself is spinning about that center of mass. Nothing new is required to handle this — you already know both kinds of kinetic energy. Rolling just means you add them:
Written out, that's K_tot = ½mv_cm² + ½Iω². There is no new physics in this equation — it's Unit 3/4 translational kinetic energy plus Unit 6.1 rotational kinetic energy, added together because both are genuinely happening at the same time.
"Rolling without slipping" means one specific physical thing: the point of the wheel touching the ground is, for that instant, not sliding against the ground. It has zero velocity relative to the surface. That single condition is powerful enough to lock the translational motion of the center of mass to the rotational motion of the whole object, through the radius r:
These three are really the same statement, just differentiated with respect to time: differentiate Δx_cm = rΔθ once and you get v_cm = rω; differentiate again and you get a_cm = rα. Learn the first one and the other two come free.
A hoop (I = mr²) and a solid disk (I = ½mr²), each of mass m and radius r, both roll without slipping at the same center-of-mass speed v_cm. Find K_tot for each in terms of m and v_cm, and compare.
Work requires a force to act through a displacement at the point where the force is applied. In ideal rolling without slipping, the contact point between the wheel and the ground has zero velocityrelative to the ground at every instant — it's momentarily at rest, even though the wheel as a whole is moving.
This is exactly the boundary statement in the CED: rolling friction— a separate, genuinely dissipative effect from deformation of the wheel or surface — is beyond the scope of AP Physics 1. Within this course, "rolling without slipping" and "no energy lost to friction" go together.
Everything in 6.5.B depended on one condition: no sliding at the contact point. Break that condition and the whole kinematic link breaks with it.
Now the contact point is sliding relative to the surface, so the force there is kinetic friction, not static friction — and kinetic friction acts through a real relative displacement. That means it dissipates energy from the system, converting mechanical energy into heat at the sliding surface.
Set v_cm and ω independently and watch the wheel roll. When v_cm = rω the contact point (marked in white) stays still relative to the ground for an instant each turn. Break that condition and the wheel slips.
The contact point is (essentially) at rest relative to the ground. v_cm = rω. Static friction does zero work here — no energy is dissipated.
Try the presets, then nudge either slider slightly away from "No Slipping" — notice v_cm − rω moves away from zero and the verdict below flips.
A wheel of radius r = 0.30 m has v_cm = 4.0 m/s and ω = 20 rad/s, spinning in the sense that would carry it forward if it were rolling without slipping. Determine whether the wheel is slipping and, if so, which kind.