Chapter 17.3, Problem 10E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# As in Exercise 9, consider a spring with mass m, spring constant k, and damping constant c = 0, and let ω = k / m . If an external force F(t) = F0 cos ωt is applied (the applied frequency equals the natural frequency), use the method of undetermined coefficients to show that the motion of the mass is given by x ( t ) = c 1 cos ω t + c 2 sin ω t + F 0 2 m ω t sin ω t

To determine

To show: If external force F(t)=F0cosωt is applied, then the motion of the mass is given by x(t)=c1cosωt+c2sinωt+F02mωtsinωt .

Explanation

Given data:

F(t)=F0cosωt , ω=km , c=0

Formula used:

Write the expression for Newton’s Second Law with external force.

md2xdt2+kx=F(t)

mx+kx=F(t) (1)

Write the expression for auxiliary equation.

mr2+k=0 (2)

Write the expression for general solution with complex roots.

x(t)=eαt[c1cos(βt)+c2sin(βt)] (3)

Write the expression for r .

r=α+βi (4)

Find the expression for Newton’s Second Law with external force using equation (1).

Substitute F0cosωt for F(t) in equation (1),

mx+kx=F0cosωt (5)

Find the roots using equation (2).

mr2=kr2=kmr=kmr=±kmi

Substitute ω for km ,

r=±ωi (6)

Compare equations (4) and (6).

α=0β=ω

Substitute 0 for α and ω for β in equation (3),

x(t)=e(0)t[c1cos(ωt)+c2sin(ωt)]

x(t)=c1cos(ωt)+c2sin(ωt) (7)

Modify equation (7) for complementary equation as follows.

xc(t)=c1cos(ωt)+c2sin(ωt)

The natural frequency of the system equals to the frequency of the external force.

Consider the value of xp(t) as follows.

xp(t)=t(Acosωt+Bsinωt) (8)

Differentiate equation (8) with respect to t .

xp(t)=t[ωAsinωt+ωBcosωt]+[Acosωt+Bsinωt]

Differentiate equation with respect to t .

xp(t)=t[ω2Acosωtω2Bsinωt]+[ωAsinωt+ωBcosωt]+[ωAsinωt+ωBcosωt]=t[ω2Acosωtω2Bsinωt]+[2ωAsinωt+2ωBcosωt]

Modify equation (5) as follows

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