Minimum descent time along a set of connected inclined planes

Abstract

The time required for a particle to slide frictionlessly down a set of ramps connected end to end can be minimized numerically as a function of the coordinates of the connection points between ramps and compared to the exact cycloidal solution of the brachistochrone problem. It is found that a set of just three joined ramps over a large range of geometrical aspect ratios has a descent time within 5% of the optimal cycloid. The special case where the particle starts and ends at the same height has sufficient symmetry that it can be analyzed analytically using algebra alone. The level of analysis is appropriate to an undergraduate classical mechanics course.

Energy, Christiaan Huygens, and the Wonderful Cycloid—Theory versus Experiment

“…The explanation for this is that when h > 30 cm, the starting points of the two rails become closer to each other, meaning that the problem becomes identical to the brachistochrone problem (the fastest path), and the cycloid pathway is the fastest out of all the possible pathways that connect the starting point with the finishing point [3]. This could also be analyzed by progressively introducing "kinks" in an inclined path, such that in the continuum limit we get a smooth curve that is the cycloid [4,5]. A comparison between the duration time achieved in the experiment with the cycloid rail and the theoretical duration time is called for.…”

“…The rotational friction coefficient µ r is usually small compared with the coefficient of kinetic friction between the surfaces (this is the great advantage inherent in the invention of the wheel). It is worth noting that the roll is caused by the static friction between the ball and the surface, but because in the nonslip rotation the point of contact between the surface and the ball is at rest relative to the surface, there is no loss of energy as a result of the static friction [2,4,6]. However, the rolling friction still works.…”

“…The said clock also needed to overcome the changes in weather and ship movements. For example, a regular pendulum clock was influenced by the changes in temperature that caused changes in the length of the string and therefore also caused changes in the pendulum's cycle times [4].…”

“…As proof that the cycloid is an isochronous pathway, he published a book entitled Horologium oscillatorium in 1673. Huygens proved this argument in an ingenious way (see Appendix A) based on basic mechanical reasoning, without using infinitesimal mathematics [2][3][4][5][6][7]. A cycloid is the curve that is created by the pathway of the point that is on the perimeter of the wheel without sliding.…”

“…A schematic description of the experimental system for measuring the duration of the ball's motion on one of the rails[4].…”