Chapter 5.1, Problem 45E

1. (a) Show that x(t) given in part (a) of Problem 43 can be written in the form

   $x(t)=\frac{-2F_0}{{\omega }^2-{\gamma }^2}\sin \frac{1}{2}(\gamma -\omega )t\sin \frac{1}{2}(\gamma +\omega )t$

2. (b) If we define $\epsilon =\frac{1}{2}(\gamma -\omega )$, show that when ε is small an approximate solution is

$x(t)=\frac{F_0}{2\epsilon \gamma }\sin \epsilon t\sin \gamma t$.

When ε is small, the frequency γ/2π of the impressed force is close to the frequency ω/2π of free vibrations. When this occurs, the motion is as indicated in Figure 5.1.23. Oscillations of this kind are called beats and are due to the fact that the frequency of sin εt is quite small in comparison to the frequency of sin γt. The dashed curves, or envelope of the graph of x(t), are obtained from the graphs of ±(F₀/2εγ) sin εt. Use a graphing utility with various values of F₀, ε, and γ to verify the graph in Figure 5.1.23.

Figure 5.1.23 Beats phenomenon in Problem 45

43.

1. (a) Show that the solution of the initial-value problem

   $\frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=F_0\cos \gamma t,x(0)=0,x^{\prime }(0)=0$

   is $x(t)=\frac{F_0}{{\omega }^2-{\gamma }^2}(\cos \gamma t-\cos \omega t)$.