To jog around the periphery of a carousel at rest requires that the jogger experience a constant state of acceleration perpendicular to the direction of motion and directed toward the axis of rotation (centripetal). The jogger could achieve this centripetal acceleration by leaning inward, thereby using a horizontal weight component to provide the necessary centripetal force. There are two ordinary cases of circular motion involving the carousel that can be handled by the simple centripetal acceleration formula, where the joggers speed v is squared and then divided by the radius r of the path being followed (a = v2/r). One case would be as above, with the carousel at rest and the jogger moving on a circular path around it. The other case would be the jogger at rest on the carousel at a radial point r while it is moving with linear speed v. The situation can be made significantly more interesting and informative by letting the jogger be in motion while the carousel is in motion.
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Here it is interesting to contemplate an imposition of radial (sideways) movements on the jogger while the forward motion remains unchanged (stationary) with respect to the non-rotating frame. Since the jogger's kinetic energy relative to the carousel must diminish with decreasing radii, and of necessity increase with increasing radii, it would take an effort (energy) to move the jogger outward, while on the other hand there would be a natural tendency for the jogger to move (spiral) inward to what would be a lower energy state, thereby giving up energy in the process. Again we see the Coriolis effect, which applies to motion in any direction on the turning carousel and which fits perfectly with the slowing down of the jogger with respect to the carousel while moving inward and the speeding up while moving outward. However, it does not absorb or contribute any of the energy that accompanies these transformations.
The use of incremental quantities is a convenient way to do derivations that become more exact in the limit Δt → 0.