Explanation Given: The cycloid represented by the parametric equations, x ( θ ) = a ( θ − sin θ ) y ( θ ) = a ( 1 − cos θ ) Formula Used: For a smooth curve given by the parametric equations x = x ( θ ) and y = y ( θ ) , the curvature is given by, k = | x ' ( θ ) ⋅ y ' ' ( θ ) − y ' ( θ ) ⋅ x ' ' ( θ ) | [ x ' ( θ ) + y ' ( θ ) 2 ] 3 2 Calculation: Consider the given parametric equations, x ( θ ) = a ( θ − sin θ ) y ( θ ) = a ( 1 − cos θ ) Differentiating the function with respect to θ , to get, x ' ( θ ) = a ( 1 − cos θ ) And, y ' ( θ ) = a sin θ Again, Differentiating the function with respect to θ , to get, x ' ' ( θ ) = a sin θ And, y ' ' ( θ ) = a cos θ Now, For a smooth curve given by the parametric equations x = x ( θ ) and y = y ( θ ) , the curvature is given by, k = | x ' ( θ ) ⋅ y ' ' ( θ ) − y ' ( θ ) ⋅

Calculus: Early Transcendental Functions

7th Edition

ISBN: 9781337552516

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 12.5, Problem 77E

To determine

To calculate: The curvature K of the cycloid represented by the parametric equations

x(θ)=a(θ−sinθ) and y(θ)=a(1−cosθ) and the minimum and maximum value of K.

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Chapter 12.5, Problem 76E

Chapter 12.5, Problem 78E

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Calculus: Early Transcendental Functions

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The curvature K of the cycloid represented by the parametric equations x ( θ ) = a ( θ − sin θ ) and y ( θ ) = a ( 1 − cos θ ) and the minimum and maximum value of K .

Solution Summary: The author calculates the curvature K of the cycloid represented by the parametric equations cx(theta -mathr

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To calculate: The curvature K of the cycloid represented by the parametric equations

x(θ)=a(θ−sinθ) and y(θ)=a(1−cosθ) and the minimum and maximum value of K.

Want to see the full answer?

Chapter 12 Solutions

The curvature K of the cycloid represented by the parametric equations x ( θ ) = a ( θ − sin θ ) and y ( θ ) = a ( 1 − cos θ ) and the minimum and maximum value of K .

Solution Summary: The author calculates the curvature K of the cycloid represented by the parametric equations cx(theta -mathr

Videos

To calculate: The curvature K of the cycloid represented by the parametric equations x(θ)=a(θ−sinθ) and y(θ)=a(1−cosθ) and the minimum and maximum value of K.

Want to see the full answer?

Chapter 12 Solutions

To calculate: The curvature K of the cycloid represented by the parametric equations

x(θ)=a(θ−sinθ) and y(θ)=a(1−cosθ) and the minimum and maximum value of K.