Damped harmonic motion Pretend we have a mass which is attached to a vertical spring. If we were to extend the spring by dragging it down and then releasing it, the weight would start to oscillate. Because of friction from  surrounding air, the amplitude will decrease and eventually reach zero. I know that the damping force can be calculate using the formula
$F = -b*v$
Now onto my question. Is it possible to calculate the constant $b$? And does this suggest that air always has the same damping constant?
 A: If the mass doesn't move too quickly, the resistive force acting on it will be due to the air's viscosity, and will be proportional to the body's velocity and in the opposite direction to it. As you say, $\mathbf F_{\text{res}}=-b\mathbf v$, in which $b$ is a constant for a particular size and shape of body and for air at a particular temperature. [For example, according to the nineteenth century Anglo-Irish physicist George Gabriel Stokes, if the body is a sphere of radius $r$, then $b=6\pi\eta r$ in which $\eta$ is the viscosity of air at the relevant temperature.] So the equation of motion of the mass is
$$m\frac{d_2x}{dt^2}=-b\frac{dx}{dt}-ax$$
in which $-ax$ is the 'restoring force' from the spring.
The solution to this equation is
$$x=Ae^{-bt/2m}\sin{(\omega t+\phi)}\ \ \ \ \ \ \text{in which}\ \ \ \ \ \ \omega=\sqrt{\frac km-\frac{b^2}{4m^2}}$$
The easiest way to determine the value of $b$ is from the damped oscillations themselves. Their initial amplitude is $A$ and their amplitude at time $t$ is $Ae^{-bt/2m}.$ Therefore the time, $t_{1/2}$, for their amplitude to fall to half its initial value is given by
$$Ae^{-(bt_{1/2}/2m)}=\tfrac A2\ \ \ \ \ \ \ \ \ \text{giving}\ \ \ \ \ \ \ \ \ b=\frac{2m\ \ln 2}{t_{1/2}}$$
So, knowing the value of the mass, $m$, we can determine $b$ by measuring the time taken for the amplitude to halve.
A: Yes you can do so, it's actually a Standard undergrad experiment to measure this dampening explicitly and determine the dampening constant. (Essentially the amplitude of the pendulum will be bounded by an exponential with exponent proportional to this constant)
However this does not mean that the air has one universal constant for this dampening. The constant is heavily dependent on the shape of the object you are looking at. Further the dampening constant depends on the density of the air and as such on external conditions such as pressure and temperature of the lab you are working in.
