   Chapter 10.2, Problem 72E

Chapter
Section
Textbook Problem

(a) Show that the curvature at each point of a straight line is κ = 0. (b) Show that the curvature at each point of a circle of radius r is κ = 1/r.

(a)

To determine

To show: The curvature of the straight line is k=0.

Explanation

Given:

Write the equation of straight line y=mx+c.

The parametric equation for the variable x is as below.

x=x

The parametric equation for the variable y is as below.

y=f(x)y=mx+c

Calculation:

Differentiate the parametric equation x with respect to θ.

x=xx˙=1

Differentiate the parametric equation y with respect to θ.

y=mx+cy˙=m

Again differentiate the parametric equation x with respect to θ.

x˙=1x¨=0

Again differentiate the parametric equation y with respect to θ.

y˙=my¨=0

The curvature formula for the curve is k=|x˙y¨y˙x¨|(x˙2+y˙2)3/2

(b)

To determine

To show: The curvature at each point of the circle of radius r is k=1r

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