   Chapter 13.3, Problem 44E

Chapter
Section
Textbook Problem

# Use the formula in Exercise 42 to find the curvature.44. x = a cos ωt, y = b sin ωt

To determine

To find: The curvature of the plane with parametric equations x=acosωt and y=bsinωt .

Explanation

Given data:

Parametric equations are x=acosωt and y=bsinωt .

Formula used:

The curvature of plane with parametric equation x=f(t) , y=g(t) is,

k(t)=|x˙y¨x¨y˙|(x˙2+y˙2)32 (1)

Here,

k(t) is curvature of the plane.

Consider the given equation.

x=acosωt

Differentiate the equation with respect to t.

Find the value of x¨ .

x¨=ddt(aωsinωt)=aω(ωcosωt) {ddx(sinax)=acosx}=aω2cosωt

Consider the given equation.

y=bsinωt

Differentiate the equation with respect to t.

y˙=ddt(bsinωt)=bddt(sinωt)=bωcosωt {ddx(sinax)=acosx}

Find the value of y¨ .

y¨=ddt(bωcosωt)=bω(ωsinωt) {ddx(cosax)=asinx}=bω2sinωt

Substitute aωsinωt for x˙ , aω2cosωt for x¨

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