   Chapter 10.2, Problem 71E

Chapter
Section
Textbook Problem

Use the formula in Exercise 69(a) to find the curvature of the cycloid x = θ − sin θ, y = 1 − cos θ at the top of one of its arches.

To determine

To find: The curvature of cycloid for the given parametric equation x=θsinθ and y=1cosθ.

Explanation

Given:

The parametric equation for the variable x is x=θsinθ.

The parametric equation for the variable y is y=1cosθ.

Calculation:

Differentiate the parametric equation x with respect to θ.

x=θsinθdxdθ=1cosθx˙=1cosθ

Differentiate the parametric equation y with respect to θ.

y=1cosθdydt=sinθy˙=sinθ

Again differentiate the parametric equation x with respect to θ.

x˙=1cosθx¨=sinθ

Again differentiate the parametric equation y with respect to θ.

y˙=sinθy¨=cosθ

The curvature formula is k=|x˙y¨y˙x¨|[x˙2+y˙2].

Substitute x¨=1cosθ, x˙=1cosθ, y˙=sinθ, y¨=cosθ in above equation k=|x˙y¨y˙x¨|[x˙2+y˙2],

k=|x˙y¨y˙x¨|[x˙2+y˙2]=(1cosθ)(cosθ)(sinθ

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