Rolling Motion

Objects like a cylinder, a sphere, or a wheel very commonly perform ROLLING motion -- and PURE rolling (rolling without any slipping at the point of contact) is genuinely a combination of two simultaneous motions happening together: (i) a purely ROTATIONAL motion of the body as a whole about its own axis of symmetry, through its centre of mass, and (ii) a purely TRANSLATIONAL motion, as if the entire mass of the body were concentrated at, and moving only as, its centre of mass. Describing the motion of any single individual particle of a rolling body (other than the centre of mass itself) directly is considerably harder -- each traces a complicated cycloid-like path -- but this rotation-plus-translation decomposition sidesteps that difficulty entirely and is exact for pure rolling.

Because the motion genuinely has both these pieces, the object correspondingly possesses BOTH kinds of kinetic energy at once -- translational (from the centre-of-mass motion) and rotational (from the spin about the centre of mass) -- and its total kinetic energy is simply their sum. For an object of mass M, radius R, moment of inertia $I=MK^2$ (K its radius of gyration) rolling uniformly with the centre-of-mass speed v (so $\omega=v/R$ for pure, non-slipping rolling), $E=\underbrace{\frac{1}{2}Mv^2}_{\text{translational}}+\underbrace{\frac{1}{2}I\omega^2}_{\text{rotational}}=\frac{1}{2}Mv^2+\frac{1}{2}(MK^2)\left(\frac{v}{R}\right)^2=\frac{1}{2}Mv^2\left(1+\frac{K^2}{R^2}\right) \qquad \text{--- (1.18)}$ It is worth being explicit that static friction (NOT kinetic/sliding friction) is essential for pure rolling -- it is precisely what prevents the point of co …