Complex representation to the rescue

(a)

Use the complex representation to find a real general solution to the equation of motion for the damped harmonic oscillator (see Problem 2). For this problem, you may assume that the drag constant $b$ is relatively small: $b <2 \sqrt{k/m}$.

Hint: To make your general solution, use $e^{\underline{\alpha}t}$ where $\underline{\alpha}$ is complex. Be sure to take the real part to get your answer. Compare your answer with the general solution in Problem 2(b) and Eq. (13-42) in YF, p. 412.

(b)

Find an expression for and sketch a graph of the particular solution, if $k = 400,000$ N/m, $m = 1000$ kg, $b = 2$ $\mathrm{s}^{-1}$, $x_{\rm eq} = 0$, $x_o = 0$, and $v_o = 10$ cm/s.