Chapter 4.3, Problem 35E

Swinging Door. The motion of a swinging door with an adjustment screw that controls the amount of friction on the hinges is governed by the initial value problem

$I{\theta }^{″}+b{\theta }^{\prime }+k\theta =0$; $\theta \left(0\right)={\theta }_0$, ${\theta }^{\prime }\left(0\right)=v_0$

where $\theta$ is the angle that the door is open, $I$ is the moment of inertia of the door about its hinges, $b>0$ is a damping constant that varies with the amount of friction on the door, $k>0$ is the spring constant associated with the swinging door, ${\theta }_0$ is the initial angle that the door is opened, and $v_0$ is the initial angular velocity imparted to the door (see Figure 4.10).If $I$ and $k$ are fixed, determine for which values of $b$ the door will not continually swing back and forth when closing.