Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section D / D12 / b / Solution

The transient is the homogeneous part of a damped harmonic oscillator, $m\ddot{x}+b\dot{x}+kx=0$. Define $\alpha =b/(2m)$ and ${\omega }_0=\sqrt{k/m}$. Its characteristic roots are $-\alpha \pm \sqrt{{\alpha }^2-{\omega }_0^2}$.

Initial position and velocity determine A,B. For an imposed force or support motion, the full solution is a particular response plus this transient, which decays for b>0 and k>0. The mechanical energy obeys $d(K+U)/dt=-b{\dot{x}}^2$. In light damping its cycle-averaged envelope decays as $e^{-2\alpha t}$, with $Q\simeq {\omega }_0/(2\alpha )$. Initial conditions can still produce an overshoot in a nonoscillatory regime; the damping category describes the roots rather than every possible trajectory.