>[!info] [Open Stax Page on this topic](https://openstax.org/books/university-physics-volume-1/pages/15-5-damped-oscillations) Introducing damping into [[Simple Harmonic Motion]] is a less idealized version of most problems where throughout time the initial [[Energy]] is transferred to other forms, such as [[Friction]] damping. With Friction being proportional to the velocity of an object (proportionality constant $\alpha$), the force of an object can be expressed as: $\huge \vec F=m\vec a = -\alpha \vec v - k\vec x$ We can rewrite this as the differential equation: $\huge \begin{align} x'' &= -\frac{b}{m} x' - \frac{k}{m} x \\ 0&= x'' + \frac{b}{m} x + \frac{k}{m} x \\ \end{align}$ Which has an analytic solution of: $\huge \begin{align} x(t) &= A_{0} e^{- \frac{b}{2m}t }\cos\pa{\omega t+\phi}\\ \omega &=\sqrt{ \frac{k}{m}- \pa{\frac{b}{2m}}^2 } \end{align} $ >[!example] >![[202509261357]]