   Chapter 17.3, Problem 11E

Chapter
Section
Textbook Problem

Show that if ω0 ≠ ω, but ω/ω0 is a rational number, then the motion described by Equation 6 is periodic.

To determine

To show: If ω0ω , but ωω0 is a rational number, then the motion described by x(t)=c1cosωt+c2sinωt+F0m(ω2ω02)cosω0t is periodic.

Explanation

Given data:

x(t)=c1cosωt+c2sinωt+F0m(ω2ω02)cosω0t , ω0ω , ωω0

Consider expression as follows.

x(t)=c1cosωt+c2sinωt+F0m(ω2ω02)cosω0t (1)

Consider the value of f(t) and g(t) as follows.

f(t)=c1cosωt+c2sinωtg(t)=F0m(ω2ω02)cosω0t

Substitute f(t) for c1cosωt+c2sinωt and g(t) for F0m(ω2ω02)cosω0t in equation (1),

x(t)=f(t)+g(t) (2)

The function f(t) is periodic with period 2πω .

If ω0ω , then the function g(t) is periodic with period 2πω0 .

Consider the rational number as ab where a and b are non-zero integers.

If ωω0 is rational number then ωω0=ab .

ωω0=aba=bωω0

Consider t is periodic with period a2πω .

Modify equation (2) as follows.

x(t+a2πω)=f(t+a2πω)+g(t+a2πω) (3)

Since, function f(t) is periodic with period 2πω , f(t+a2πω) is same as f(t) .

Substitute f(t) for f(t+a2πω) in equation (3),

x(t+a2πω)=f(t)+g(t+a2πω) (4)

Find the value of g(t+a2πω)

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