   Chapter 13.2, Problem 52E

Chapter
Section
Textbook Problem

# If r is the vector function in Exercise 51, show that r''(t) + ω2r(t) = 0.

To determine

To show: The expression r(t)+ω2r(t) is equals to 0 when the vector r(t)=acosωt+bsinωt where a and b are constant vectors.

Explanation

Calculation of vector r(t) :

Differentiate each component of the vector function r(t)=acosωt+bsinωt to obtain the vector r(t) as follows.

ddt[r(t)]=ddt(acosωt+bsinωt)

As a and b are constant vectors, rewrite the expression as follows.

Use the following formulae and compute the expression.

ddtcosωt=ωsinωtddtsinωt=ωcosωt

Compute the expression r(t)=addt(cosωt)+bddt(sinωt) as follows.

r(t)=a(ωsinωt)+b(ωcosωt)

r(t)=aωsinωt+bωcosωt (1)

Calculation of vector r(t) :

Differentiate equation (1) with respect to t.

ddt[r(t)]=ddt(aωsinωt+bωcosωt)

As a and b are constant vectors, rewrite and compute the expression as follows

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