Rolling

Rolling is a type of rigid-body motion in which a body turns about an axis while its point of contact with a surface instantaneous has zero velocity relative to the surface. For rolling without slipping, the motion is a combination of rotation and translation such that the contact point does not slide. Rolling is common for wheels, cylinders, and spheres.

In the simplest case, a wheel of radius R translates with speed v and rotates with angular speed ω, related by v = ωR. The kinetic energy is the sum of translational and rotational: KE = 1/2 m v^2 + 1/2 I ω^2. For rolling without slipping, ω = v/R, so KE = 1/2 (m + I/R^2) v^2. For a solid disk I = 1/2 mR^2, leading to KE = 3/4 m v^2.

Friction plays a key role: static friction enables rolling without slipping; if friction is insufficient, slipping occurs, producing rolling friction or kinetic energy loss. Rolling resistance arises from deformation of the body and surface, air drag, and micro-slip at the contact patch.

Applications include transportation (vehicle wheels and bicycles), machinery (rollers in conveyors and printing presses), and sports equipment (ball games rely on rolling contact). In railways, wheels roll along tracks; rolling is fundamental to efficient motion.

In materials engineering, rolling refers to a metal-forming process in which a billet or slab is passed between rollers to reduce thickness or alter cross-section. Hot rolling and cold rolling produce products such as rolled steel, sheet, and rail shapes.