   Chapter 17.3, Problem 9E

Chapter
Section
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)=F0cosω0t , ω=km , ω0ω .

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 F0cosω0t for F(t) in equation (1),

mx+kx=F0cosω0t (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)

Consider the value of xp(t) as follows.

xp(t)=Acosω0t+Bsinω0t (8)

Differentiate equation (8) with respect to t .

xp(t)=ω0Asinω0t+ω0Bcosω0t

Differentiate equation with respect to t .

xp(t)=ω02Acosω0tω02Bsinω0t

Modify equation (5) as follows

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