The damped harmonic oscillator is an extension of the previous discussion by the addition of another force term to model damping.

Clearly identify “the forces”, the restoring force, \[F\left( x \right)=-m\omega _{0}^{2}x\] and and the damping force, \[F_{d}\left( x \right)=-b \dot{x}\]. Set the sum equal to the acceleration, \(a=\frac{d^{2}x}{dt^{2}}\). (Check that the \(\dot{x}\) notation is familiar).

Discuss other types of damping forces briefly. Also metion overdamped and critically damped cases, but focus on underdamped case.

Solve the differential equation postulating a solution \(Ce^{pt}\), and obtain a solution in the “C-form” with \(e^{\pm i\left( \omega _{0}+i\beta \right)t}\) terms. Find exponential damping, discuss limits etc.

Class discussion about relationship to the pendulum laboratory. Is this a good model of that damping?