rollingball

Rolling ball refers to the motion of a spherical body that moves over a surface by rotating about its axis, typically in a regime called pure rolling where the point of contact is momentarily at rest. In pure rolling, the translational speed of the ball’s center of mass v and its angular speed ω are tied by the relation v = ω r, with r the ball’s radius. The contact geometry makes the instantaneous contact point the center of zero velocity, while the rest of the ball rolls forward.

Friction plays a key role. Static friction provides the torque that enables rotation without slipping. If the surface is too rough or too smooth, or if external forces exceed static friction, the ball may slip, breaking the v = ω r relation and altering the motion.

Kinetic energy in rolling motion combines translation and rotation. For a solid sphere, the total kinetic energy is (1/2) m v^2 + (1/2) I ω^2, with I = (2/5) m r^2. Using v = ω r, this becomes (7/10) m v^2. The distribution of rotational inertia depends on shape; for a hollow sphere I = (2/3) m r^2, leading to different energy sharing and dynamics.

On an incline, a common result for rolling without slipping is a = g sin(θ) / (1 + I/(m r^2)). For a solid sphere, a = (5/7) g sin(θ); for a hollow sphere, a = (2/3) g sin(θ). Applications include wheels and ball bearings in machinery, sports equipment such as bowling balls and soccer balls, and roly-poly toys. Rolling resistance, material deformation, and surface texture influence real-world behavior.

See also: rolling motion, rotational dynamics, moment of inertia, rolling without slipping.