Cycloid

Definition

A cycloid is a curve traced by a point on the circumference of a circle as it rolls along a straight line. It is a roulette curve, meaning it is generated by the motion of one curve rolling on another.

5 Must Know Facts For Your Next Test

1. The equation of a cycloid in parametric form is $x(t) = a(t - \sin t)$ and $y(t) = a(1 - \cos t)$, where $a$ is the radius of the generating circle.
2. The cycloid has a cusp at the points where the generating circle touches the straight line, and the curve is symmetric about a vertical line through the cusps.
3. The length of one complete cycle of a cycloid is $8a$, where $a$ is the radius of the generating circle.
4. The area enclosed by one complete cycle of a cycloid is $3\pi a^2$, where $a$ is the radius of the generating circle.
5. Cycloids have many interesting properties, such as the fact that the path of a bead sliding without friction on an inverted cycloid is a tautochrone, meaning the time it takes the bead to reach the bottom is independent of its starting position.

Review Questions

• Explain how the cycloid is related to parametric equations and describe the general form of the parametric equations for a cycloid.
  • The cycloid is a curve that can be represented using parametric equations, which define the $x$ and $y$ coordinates of the curve as functions of a parameter, typically denoted as $t$. The general form of the parametric equations for a cycloid are $x(t) = a(t - \sin t)$ and $y(t) = a(1 - \cos t)$, where $a$ is the radius of the generating circle. These parametric equations capture the motion of a point on the circumference of a circle as it rolls along a straight line, which is the defining characteristic of a cycloid.
• Describe the key properties of a cycloid, including its symmetry, the length of one complete cycle, and the area enclosed by one cycle.
  • The cycloid has several important properties. First, it is symmetric about a vertical line through the cusps, where the generating circle touches the straight line. Second, the length of one complete cycle of a cycloid is $8a$, where $a$ is the radius of the generating circle. Third, the area enclosed by one complete cycle of a cycloid is $3\pi a^2$, where $a$ is the radius of the generating circle. These properties are crucial for understanding the behavior and applications of cycloids, particularly in the context of parametric equations and their graphical representations.
• Explain the connection between cycloids and the concept of tautochrones, and discuss the significance of this relationship.
  • Cycloids have a unique property related to the concept of tautochrones, which are curves along which the time taken by an object to slide down the curve is independent of its starting position. Specifically, the path of a bead sliding without friction on an inverted cycloid is a tautochrone. This means that the time it takes the bead to reach the bottom of the curve is the same, regardless of where it starts. This property of cycloids has important applications in the design of mechanical systems, such as clocks and pendulums, where maintaining a constant period of oscillation is crucial.

Related terms

Parametric Equations:

Parametric equations are a way of representing a curve or surface in the coordinate plane or space by using a parameter, typically denoted by $t$, to define the $x$ and $y$ (and sometimes $z$) coordinates.

Roulette Curve: A roulette curve is a plane curve generated by the motion of a point on the circumference of a circle as it rolls along a straight line without slipping.

Trochoid: A trochoid is a curve generated by a point on the circumference of a circle as the circle rolls along a straight line. The cycloid is a special case of the trochoid where the generating circle has a radius equal to the distance it rolls.