Question

Find the length l of one "arch" of this cycloid, that is, find the distance traveled by a small stone stuck in the tread of a tire of radius r during one revolution of the rolling tire.

Answer 1

Answer: L = 2*R*π

Explanation: Considering the case where no other forces or effects are happening (Sliding, deformation, etc) The path that the rock does inside of a circular object is equal to the perimeter of the circular object, in this case, a rolling tire.

Then if the radius of the tire is R; the perimeter of the tire is equal to:

p = 2*R*π  

And for one revolution of the tire, the length of the arc that the rock travels is equal to the perimeter; so we have:

L = P = 2*R*π

Answer 2

If there is no slipping or skidding going on, then for each revolution, the tire rolls along the ground a distance equal to the tire's circumference.

Circumference of the tire = 2 · π · Radius .

During the same revolution, the small stone stuck in the tread travels in a big semicircular arc that's  (2 · π · r) long.  (roughly 6.28r)