2. Derivation for a spring-mass system with damping
Now consider the same scenario, where the effect of air resistance is not neglected in the motion. The viscous medium provides a drag force proportional to the velocity, \(F_{drag\ }=\ -bv\).
Initial conditions: Assume that at \(t=0\), we have \(v=0\) and \(x=A>0\).
2.1 Write down the equation of motion of the mass in terms of \(x\) and its time derivatives. Arrange it in a form \(\ddot{x} + 2 \gamma \dot{x} + \omega^2 x = 0\). Find expressions for \(\gamma\) and \(\omega\). Note that both \(\gamma\) and \(\omega\) have dimensions of inverse time.