Mechanical engineers often study the movement of damped oscillating objects. Such studies play an important role in the design of automobile suspensions, among other things.
The horizontal displacement of an object is given as a function of time by the equation
$x = x_0 e^{-\beta t} [\cos(\omega t) + (\beta/\omega)\sin(\omega t)]$
The parameters $\beta$ and $\omega$ depend upon the mass of the object and the dynamic characteristics of the system (i.e., the spring constant and the damping constant).
For a given system, suppose $x_0 = 8$ in, $\beta = 0.1$ s$^{-1}$, and $\omega = 0.5$ s$^{-1}$.
Carry out the following calculations for this system:
(a) Plot $x$ versus $t$ over the interval $0 \le t \le 30$ seconds.
(b) Determine, as accurately as possible, the 5 points where $x$ crosses the $t$-axis (i.e., determine the value of $t$ corresponding to $x = 0$).
(c) Determine, as accurately as possible, the maximum value between 10 and 15 seconds and the minimum value between 15 and 25 seconds.
(d) If $x_0$ and $\beta$ retain their original values, what value of $\omega$ will cause $x$ to cross the $t$-axis for the first time at $t = 2.5$ seconds?
Part (d) is for extra credit.
