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

$\mathbf{I\theta }\text{'}\text{'}\mathbf{+}\mathbf{b\theta }\text{'}\mathbf{+}\mathbf{k\theta }\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{\theta }\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{\theta }}_0\mathbf{,}\mathbf{\theta }\text{'}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{v}}_0$,

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

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