(a) Show that x(t) given in part (a) of can be written in the form -2Fo 1 1 x(t) = 2 2

Chapter 3, Problem 41

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(a) Show that x(t) given in part (a) of can be written in the form -2Fo 1 1 x(t) = 2 2 sin -('Y - w)t sin-('Y + w)t. w - 'Y 2 2 (b) If we defme e = !< 'Y - w), show that when e is small, an approximate solution is Fo . . x(t) = -smet sm 'Yt. 2e'Y When e is small the frequency 'Yl2'TT of the impressed force is close to the frequency wl2'TT of free vibrations. When this occurs, the motion is as indicated in FIGURE 3.8.22. Oscillations of this kind are called beats and are due to the fact that the frequency of sin et is quite small in comparison to the frequency of sin 'Yt. The red dashed curves, or envelope of the graph of x(t), are obtained from the graphs of (F0/2e'Y) sin et. Use a graphing utility with various values of F0, e, and 'Y to verify the graph in Figure 3.8.22. x FIGURE 3.8.22 Beats phenomenon in 41