Press the button: the chain-angle equation is checked and independently integrated as a damped pendulum, and the rotating-frame ball path is integrated with RK4 and compared with the straight ground-frame line.

Seen from the ground a free ball moves in a straight line (Newton's first law). Seen from a rider on the turning disc it follows a curve, because the disc frame is not inertial. With disc angular velocity w the apparent accelerations in the rotating frame are the centrifugal term w^2 r (outward) and the Coriolis term -2 w x v (sideways, proportional to speed). The orange curve is the ball's path as the rider sees it, the blue dashed line is the straight path from the ground.

Swing-ride chair: a chain of length L hangs from an arm of radius r0. In steady rotation the chair sits at radius r = r0 + L sin(theta) and the balance of gravity and the needed centripetal force gives tan(theta) = w^2 (r0 + L sin theta) / g, solved here by fixed-point iteration. Tension per kilogram is g / cos(theta); apparent weight is sqrt(1 + (w^2 r / g)^2) in g. The chain angle does not depend on the rider's mass.

Simplifications: rigid point rider, no air drag, no friction in the swing bearing (a small damping is used only so the simulated chair settles). Same physics as the 3D page.