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In this problem, we will be thinking about a parametric curve and imagining a string unwound from that curve. We are trying to think about the path traced out by the end of the string as if there was a pen on that string. Here is an animation of this happening for a few different curves. The red curve is the one from which the string is unwound, the black dotted line represents the string, and the green curve represents the curve traced out by the end of that string. This green traced curve is the one we'll be interested in describing.

In this problem, we will be thinking about a parametric curve and imagining a string unwound from that curve. We are trying to think about the path traced out by the end of the string as if there was a pen on that string. Here is an animation of this happening for a few different curves. The red curve is the one from which the string is unwound, the black dotted line represents the string, and the green curve represents the curve traced out by the end of that string. This green traced curve is the one we'll be interested in describing.

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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Unwinding String In this problem, we will be thinking about a parametric curve and imagining a string unwound from that curve. We are trying to think about the path traced out by the end of the string as if there was a pen on that string. Here is an animation of this happening for a few different curves. The red curve is the one from which the string is unwound, the black dotted line represents the string, and the green curve represents the curve traced out by the end of that string. This green traced curve is the one we'll be interested in describing. Preliminary problems Draw a line with positive slope m. Label the acute angle this line makes with the x-axis as theta. Find a relationship between m and theta. Draw a line with negative slope m. Label the acute angle this makes with the x-axis as theta. Show that the same relationship from above holds. Curve 1 Consider the curve described by the parametric equations X(t) = 1³13, animation, by the way) Find the arc length along this curve from the origin (t=0) to point T_0 (t= t_0), some arbitrary point. Y ● Stang {X(+) = (1/3 (y() = 4²/2 Exton 1/3 X To make the rest of the computations easier, assume that the string initially extends a distance of 1/3 beyond the origin. What is the length of the string from the point T_0 to the point P? and y(t) = 1²12, (this is the middle {X(+) = 1/3 7(4) = 1²/2
Transcribed Image Text:Unwinding String In this problem, we will be thinking about a parametric curve and imagining a string unwound from that curve. We are trying to think about the path traced out by the end of the string as if there was a pen on that string. Here is an animation of this happening for a few different curves. The red curve is the one from which the string is unwound, the black dotted line represents the string, and the green curve represents the curve traced out by the end of that string. This green traced curve is the one we'll be interested in describing. Preliminary problems Draw a line with positive slope m. Label the acute angle this line makes with the x-axis as theta. Find a relationship between m and theta. Draw a line with negative slope m. Label the acute angle this makes with the x-axis as theta. Show that the same relationship from above holds. Curve 1 Consider the curve described by the parametric equations X(t) = 1³13, animation, by the way) Find the arc length along this curve from the origin (t=0) to point T_0 (t= t_0), some arbitrary point. Y ● Stang {X(+) = (1/3 (y() = 4²/2 Exton 1/3 X To make the rest of the computations easier, assume that the string initially extends a distance of 1/3 beyond the origin. What is the length of the string from the point T_0 to the point P? and y(t) = 1²12, (this is the middle {X(+) = 1/3 7(4) = 1²/2
At T_0, the string is tangent to the curve. Find the slope of this tangent line. Y To String r -path trand by P tagent line tament line Draw a right triangle that shows the relationship between P and T_0. Which angle is the same as the angle that this line makes with the x axis? Label it theta. -useful tringle -path traced by P {X(+) = 1/3 7(4) = 1²/2 {X(+)=t-sint {y(t) = 1-cast {X(+) = 1/3 Y(4) = 1²/2 X ● Use the slope of this tangent line, the relationship you described in the Preliminary problems, and the triangle from above to describe the coordinates of point P in terms of the parameter t_0. Eliminate the parameter of your result, and identify the kind of curve that is traced by the point P. Screen Shot 201 d... Curve 2 - Consider the cycloid described by X(t) = t - sin(t); y(t) = 1 − cos(t). This time, we will unroll the string from the right side of this curve where P starts at the top of the cycloid. Petto Pet=R то Repeat the steps above to show that the curve traced by point p is also a cycloid, just shifted. Along the way, be sure to clearly illustrate your computations with pictures. You may need the trig identity - cos α = = 2 sin² (α/2). Once you finish your parametrization of the path P traces, use integrals to describe the area bounded between the path of P and the original red curve. What is the widest interval for the limits of integration for this computation to make sense?
Transcribed Image Text:At T_0, the string is tangent to the curve. Find the slope of this tangent line. Y To String r -path trand by P tagent line tament line Draw a right triangle that shows the relationship between P and T_0. Which angle is the same as the angle that this line makes with the x axis? Label it theta. -useful tringle -path traced by P {X(+) = 1/3 7(4) = 1²/2 {X(+)=t-sint {y(t) = 1-cast {X(+) = 1/3 Y(4) = 1²/2 X ● Use the slope of this tangent line, the relationship you described in the Preliminary problems, and the triangle from above to describe the coordinates of point P in terms of the parameter t_0. Eliminate the parameter of your result, and identify the kind of curve that is traced by the point P. Screen Shot 201 d... Curve 2 - Consider the cycloid described by X(t) = t - sin(t); y(t) = 1 − cos(t). This time, we will unroll the string from the right side of this curve where P starts at the top of the cycloid. Petto Pet=R то Repeat the steps above to show that the curve traced by point p is also a cycloid, just shifted. Along the way, be sure to clearly illustrate your computations with pictures. You may need the trig identity - cos α = = 2 sin² (α/2). Once you finish your parametrization of the path P traces, use integrals to describe the area bounded between the path of P and the original red curve. What is the widest interval for the limits of integration for this computation to make sense?
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