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.

The length L of one “arch” of the cycloid of the stone which stuck in the tread of a tire is 2πr.

The length of arc is the measure of the length of two points of a section of a curve.

In the given problem, the stone is stuck in the tread of a tire. Now this stone will move the same distance as the one surface point of the tire in one revolution of the rolling tire.

The radius of the tire is r. Now the, distance traveled by object in one revolution of the rolling tire is equal to the circumference of the tire.

The circumference of the tire with radius r is,

As the length of the arch of this stone is the same as the circumference of the tire. Therefore, the length of the arch of stone is,

Hence, the length L of one “arch” of the cycloid of the stone which stuck in the tread of a tire is 2πr.

Learn more about the arc length here:

link

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