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The parametric equations describing the path of a point on a circle of radius r as it rolls around the inside of a circle of radius R are x(0) = (R-r) cos(0) + r cos(( − 1) 0) y(0) = (R-r) sin(0) — r sin(( − 1) 0)

The parametric equations describing the path of a point on a circle of radius r as it rolls around the inside of a circle of radius R are x(0) = (R-r) cos(0) + r cos(( − 1) 0) y(0) = (R-r) sin(0) — r sin(( − 1) 0)

Advanced Engineering Mathematics
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 need question 4,5,6,7

Rolling Circles The parametric equations describing the path of a point on a circle of radius r as it rolls around the inside of a circle of radius R are [ x(0) = (R-r) cos(0) + r cos((-1) 0) y(0) = (R-r) sin(0) - r sin((-1) 0) See ::This Desmos Graph:: for an animation 1. What path does the point trace out if R/r = 4? Do not directly plot this graph, instead give a geometric argument for why you think so. 2. As a preliminary step to the next question, prove the identities COS(30) = 4 cos³ (0) — 3 cos(0) and sin(30) = 3 sin(0) - 4 sin³ (0). O O 3. Let R=1 and r = 1/4. Use these values in the parametric equation above, then eliminate the parameter to get a nice equation. Your final answer shouldn't involve inverse trigonometry. Hint: try thinking about the trigonometric pythagorean identity. 4. Now use the more general relationship R/r = 4 (or equivalently r = R/4). Eliminate the parameter to derive the general formula of this curve. O Hint: Use the sum of angles formulas sin(a+b) = sin(a)cos(b) + sin(b)cos(a) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b) as your first step. Hint: 3theta = 2theta + theta. O 5. You may conjecture that if R/r = n for some whole-number ratio, then maybe the path traced out can be described by an equation of the form x + y = Rk. Give a simple argument about why this cannot be the correct equation. O (What I mean by "simple argument" is that it does not take lots of computations, just some simple observations about algebra and geometry. Finding the argument itself may not necessarily be easy). 6. Can we find a simple form for the equation describing the path traced out if R/r = 2? What kind of curve is this?
Transcribed Image Text:Rolling Circles The parametric equations describing the path of a point on a circle of radius r as it rolls around the inside of a circle of radius R are [ x(0) = (R-r) cos(0) + r cos((-1) 0) y(0) = (R-r) sin(0) - r sin((-1) 0) See ::This Desmos Graph:: for an animation 1. What path does the point trace out if R/r = 4? Do not directly plot this graph, instead give a geometric argument for why you think so. 2. As a preliminary step to the next question, prove the identities COS(30) = 4 cos³ (0) — 3 cos(0) and sin(30) = 3 sin(0) - 4 sin³ (0). O O 3. Let R=1 and r = 1/4. Use these values in the parametric equation above, then eliminate the parameter to get a nice equation. Your final answer shouldn't involve inverse trigonometry. Hint: try thinking about the trigonometric pythagorean identity. 4. Now use the more general relationship R/r = 4 (or equivalently r = R/4). Eliminate the parameter to derive the general formula of this curve. O Hint: Use the sum of angles formulas sin(a+b) = sin(a)cos(b) + sin(b)cos(a) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b) as your first step. Hint: 3theta = 2theta + theta. O 5. You may conjecture that if R/r = n for some whole-number ratio, then maybe the path traced out can be described by an equation of the form x + y = Rk. Give a simple argument about why this cannot be the correct equation. O (What I mean by "simple argument" is that it does not take lots of computations, just some simple observations about algebra and geometry. Finding the argument itself may not necessarily be easy). 6. Can we find a simple form for the equation describing the path traced out if R/r = 2? What kind of curve is this?
7. If R/r = 4, use the parametric equations to find the arc length traveled along the curve as a function of theta. Then, use this to reparameterize your curve with respect to arc length. O You can restrict your attention to the first quadrant for simplicity, but make note of where this assumption affects your computations.
Transcribed Image Text:7. If R/r = 4, use the parametric equations to find the arc length traveled along the curve as a function of theta. Then, use this to reparameterize your curve with respect to arc length. O You can restrict your attention to the first quadrant for simplicity, but make note of where this assumption affects your computations.
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