# Damped oscillations

Suppose I have an oscillating system, that oscillates under a standard restoring force given by :

$F=-kx,$

where $$x$$ is the displacement and $$k$$ is the constant of proportionality with $$k > 0$$.

Now suppose it is being damped by a viscous force given by the standard equation of

$-b\dot { x } ={ F }_{ drag }.$

Suppose that the particle under the forces have mass $$m$$, and at $$t = 0$$ we have $$x={ x }_{ 0 }$$.
Then what is the limiting value of $$b$$ for which the particle will not cross the origin even once?

Details and Assumptions

• The graph of the particle will appear as shown.
• Only consider the positive side, x-axis is time and y-axis displacement
• m=1 kg
• K=1 N/m
• No driving force acts on the object.
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