Chapter 17.3, Problem 9E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Suppose a spring has mass m and spring constant k and let ω = k / m . Suppose that the damping constant is so small that the damping force is negligible. If an external force F(t) = F0 cos ω0t is applied, where ω0 ≠ ω, use the method of undetermined coefficients to show that the motion of the mass is described by Equation 6.

To determine

To show: If external force F(t)=F0cosω0t is applied, where ω0ω , then the motion of the mass is described by x(t)=c1cosωt+c2sinωt+F0m(ω2ω02)cosω0t .

Explanation

Given data:

F(t)=F0cosÏ‰0t , Ï‰=km , Ï‰0â‰ Ï‰ .

Formula used:

Write the expression for Newtonâ€™s Second Law with external force.

md2xdt2+kx=F(t)

mxâ€³+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)=eÎ±t[c1cos(Î²t)+c2sin(Î²t)] (3)

Write the expression for r .

r=Î±+Î²i (4)

Find the expression for Newtonâ€™s Second Law with external force using equation (1).

Substitute F0cosÏ‰0t for F(t) in equation (1),

mxâ€³+kx=F0cosÏ‰0t (5)

Find the roots using equation (2).

mr2=âˆ’kr2=âˆ’kmr=âˆ’kmr=Â±kmi

Substitute Ï‰ for km ,

r=Â±Ï‰i (6)

Compare equations (4) and (6).

Î±=0Î²=Ï‰

Substitute 0 for Î± and Ï‰ for Î² in equation (3),

x(t)=e(0)t[c1cos(Ï‰t)+c2sin(Ï‰t)]

x(t)=c1cos(Ï‰t)+c2sin(Ï‰t) (7)

Modify equation (7) for complementary equation as follows.

xc(t)=c1cos(Ï‰t)+c2sin(Ï‰t)

Consider the value of xp(t) as follows.

xp(t)=Acosâ€‰Ï‰0t+Bsinâ€‰Ï‰0t (8)

Differentiate equation (8) with respect to t .

xâ€²p(t)=âˆ’Ï‰0Asinâ€‰Ï‰0t+Ï‰0Bcosâ€‰Ï‰0t

Differentiate equation with respect to t .

xâ€³p(t)=âˆ’Ï‰02Acosâ€‰Ï‰0tâˆ’Ï‰02Bsinâ€‰Ï‰0t

Modify equation (5) as follows

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