"Suppose we have a mass-spring-dashpot system with $m=10$, $c=9$, and $k=2$, with $x(0) = 0$ and $x'(0) = 5.$ Find how far to the right this mass moves before starting back towards the origin."
*Note: $c$ is the resistance of the medium (dashpot), $k$ is the spring constant.
My Work:
I think I'm misunderstanding this concept, so I'll outline my thought process. The test to determine whether this system is over, under, or critically damped is:
$c^2 > 4km \to$ overdamped
$c^2 = 4km \to$ critically damped
$c^2 < 4km \to$ underdamped
In this case, $c^2 = 9^2 = 81, 4km = 4\cdot2\cdot10 = 80.$ Therefore, $c^2 > 4km $ and the system should be overdamped, meaning there is no oscillation and the mass will never move back towards the origin. However, according to my textbook, the answer is actually 4.1, meaning that my logic is wrong here, and the system must be underdamped. Where did I go wrong to arrive at my conclusion?
The error in your logic is that an overdamped system may actually cross the origin. In fact, you can show that such a system may cross the origin at most once.