I am currently working on this book: https://openstax.org/books/university-physics-volume-1/pages/15-5-damped-oscillations
The position as a function of time for damped oscillations is given by $$x(t)=A_0e^{-\frac{bt}{2m}}\cos(\omega t+\phi).$$
It is basically the equation for SHM but also including an exponential decay rate and I would like to know how it is found.
Best Answer
The figure 15.26 in your reference shows the exponential damping factor as the red dashed curve. If you take the natural logarithm of this curve you get a linear function, the slope of which gives you the decay rate $b/2m$.