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A circle C of radius 2r has its center at the origin. A circle of radius r rolls without slipping in the counterclockwise direction around C. A point P is located on a fixed radius of the rolling circle at a distance b from its center, 0

A circle C of radius 2r has its center at the origin. A circle of radius r rolls without slipping in the counterclockwise direction around C. A point P is located on a fixed radius of the rolling circle at a distance b from its center, 0

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter10: Analytic Geometry
Section10.4: Analytic Proofs
Problem 32E
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A circle C of radius 2r has its center at the origin. A circle of radius r rolls without slipping in the counterclockwise direction around C. A point P is located on a fixed radius of the rolling circle at a distance b from its center, 0<b<r. Let L be the line from the center of C to the center of the rolling circite and be the angle that L makes with the positive x-axis. Using t as a parameter we will see in a video (coming soon) that parametric equations of the path traced out by P are x(t) = 3r cost + b cos 3t and y(t) = 3r sint + b sin 3t. In most rotary engines the sides of the equilateral triangle (the rotor) are replaced by arcs of circles centered at the opposite vertices, thus the diameter of the rotor is constant. Show that the rotor will fit in the epitrochoid if b ≤ 3/2 (2 - √3)r
Transcribed Image Text:A circle C of radius 2r has its center at the origin. A circle of radius r rolls without slipping in the counterclockwise direction around C. A point P is located on a fixed radius of the rolling circle at a distance b from its center, 0<b<r. Let L be the line from the center of C to the center of the rolling circite and be the angle that L makes with the positive x-axis. Using t as a parameter we will see in a video (coming soon) that parametric equations of the path traced out by P are x(t) = 3r cost + b cos 3t and y(t) = 3r sint + b sin 3t. In most rotary engines the sides of the equilateral triangle (the rotor) are replaced by arcs of circles centered at the opposite vertices, thus the diameter of the rotor is constant. Show that the rotor will fit in the epitrochoid if b ≤ 3/2 (2 - √3)r
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