The cycloid is the curve traced by a point on the rim of a rolling wheel. Imagine a wire shaped like an inverted cycloid (see figure). A bead sliding down this wire without friction has some remarkable properties. Among all wire shapes, the cycloid is the shape that produces the fastest descent time (see the Guided Project The amazing cycloid for more about the brachistochrone property). It can be shown that the descent time between any two points 0 ≤ a < b ≤ π on the curve is

where g is the acceleration due to gravity, t = 0 corresponds to the top of the wire, and t = π corresponds to the lowest point on the wire.

a. Find the descent time on the interval [a, b] by making the substitution u = cos t.
b. Show that when b = π, the descent time is the same for all values of a; that is, the descent time to the bottom of the wire is the same for all starting points.

Transcribed Image Text:

dt, cos t V g(cos e descent time = – cos t) top of wire t = 0 t = a lowest point on wire t = b +