04 / CHAPTER 06
Rolling without slipping
Combine translation and rotation through a geometric constraint.
ANCHOR EQUATION
Rolling constraint
In symbols: vcm = Rω
No slip ties translation to rotation at the contact point.
Rolling couples translation to rotation
A rolling body advances because its point of contact is instantaneously at rest relative to the surface. This is a geometric constraint, not a universal consequence of friction; slipping changes it.
Differentiate the no-slip condition
For a wheel of radius R, distance traveled is x = Rθ when rolling without slip. Differentiating gives v_cm = Rω and a_cm = Rα. The total kinetic energy is translational plus rotational: K = ½Mv² + ½Iω² = ½(M + I/R²)v².
A solid disk has I = ½MR². Rolling down a height h without losses gives Mgh = ½Mv² + ½(½MR²)(v²/R²) = ¾Mv², so v = √(4gh/3). From h = 1 m, v = 3.62 m/s.
Friction can be static and still do no work
The contact point has zero displacement in ideal rolling, so static friction may supply torque without dissipating energy. If the wheel slips, kinetic friction does work and v ≠ Rω.
Work it: Compare the final speeds of a hoop (I = MR²) and a solid cylinder (I = ½MR²) rolling from h = 0.8 m. Which arrives first and why?