A circle of radius 1 rolls along a coordinate line in the...

A circle of radius 1 rolls along a coordinate line in the positive direction, as shown in the figure. If point $P$ is initially at the origin, find the coordinate of $P$ after one, two, and ten complete revolutions.
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Key Concepts

Parametric Representation of Motion

Parametric equations are used to model the path of the rolling point by expressing its coordinates in terms of a parameter, often the angle of rotation. In the case of a cycloid, these equations combine the circle’s rotational factors (sine and cosine functions) with the linear translation and allow determination of the point's exact coordinates after any number of rotations.

Cycloid

A cycloid is the curve traced by a fixed point on the circumference of a circle as it rolls along a straight line. Understanding the cycloid involves recognizing how the point’s motion combines both rotational and translational movement, which is key to determining its position after a given number of rotations.

Rolling Without Slipping

Rolling without slipping is a condition where the circle rotates such that its point of contact with the line does not slide. This means the distance the circle travels horizontally in one full revolution is equal to its circumference. This principle is essential for relating the number of revolutions to the horizontal displacement of the tracing point.

Circumference of a Circle

The circumference of a circle, given by the formula C = 2?r, is crucial in this context because it represents the distance the circle travels in one complete revolution. For a circle with radius 1, the circumference is 2?, directly linking the number of revolutions to the distance moved along the coordinate line.

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