Question

A marble is rolled on the floor in a straight line from one person to another person. The marble has a radius of 0.6 cm and there is a fixed point Plocated on themarble. Let the person rolling the marble represent the origin. Find parametric equations in terms of O describing the cycloid traced out by P.

Answer 1

SOLUTION:

Case: Cycloid

Cycloid, the curve generated by a point on the circumference of a circle that rolls along a straight line. If r is the radius of the circle and θ (theta) is the angular displacement of the circle, then the polar equations of the curve are

x = r(θ - sin θ) and y = r(1 - cos θ)

Given:

radius, r= 0.6cm

Required: Get equations for x and y

Method:

By substitution,

x = r(θ - sin θ)

x = 0.6(θ - sin θ)

x= 0.6θ - 0.6sin θ

AND

y = r(1 - cos θ)

y = 0.6(1 - cos θ)

y= 0.6 - 0.6cos θ

Final answer:

x(θ)= 0.6θ - 0.6sin θ

AND

y(θ)= 0.6 - 0.6cos θ