2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius b centered at the origin.
2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius b centered at the origin.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Just #2. Thank you.
![**Problems 2–3.** 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 circle and \( t \) 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
\[ x(t) = 3r \cos t + b \cos 3t \quad \text{and} \quad y(t) = 3r \sin t + b \sin 3t \]
**Note:** The path traced out by \( P \) is called an **epitrochoid**.
2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius \( b \) centered at the origin.
**Note:** This is the principle of the Wankel rotary engine. When the equilateral triangle rotates with its vertices on the epitrochoid, its centroid sweeps out a circle whose center is at the center of the curve.
3. In most rotary engines the sides of the equilateral triangles 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 \( b \leq \frac{3}{2}(2 - \sqrt{3})r \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F1fd48c88-cf6f-4417-a325-c7d7feb78505%2Ft37nf6n_processed.png&w=3840&q=75)
Transcribed Image Text:**Problems 2–3.** 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 circle and \( t \) 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
\[ x(t) = 3r \cos t + b \cos 3t \quad \text{and} \quad y(t) = 3r \sin t + b \sin 3t \]
**Note:** The path traced out by \( P \) is called an **epitrochoid**.
2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius \( b \) centered at the origin.
**Note:** This is the principle of the Wankel rotary engine. When the equilateral triangle rotates with its vertices on the epitrochoid, its centroid sweeps out a circle whose center is at the center of the curve.
3. In most rotary engines the sides of the equilateral triangles 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 \( b \leq \frac{3}{2}(2 - \sqrt{3})r \).
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