What is the damping coefficient of air? How about steel?

Question

I need the value for the damping coefficient of air for a mass-spring system simple harmonic motion experiment. I can't seem to find a value for this anywhere else.

Answer 1

If the Reynolds number is confortably larger than one ($\mathit{Re} = \frac{U L}{\nu} \gg 1$, where $U$ is the velocity of the mass, $L$ its diameter and $\nu$ the kinematic viscosity of air), which I think is the most likely scenario, you can use an empirical drag coefficient to calculate the air resistance, which is proportional to the velocity squared.

In fact, I believe Newton himself was able to give the first estimation of the drag coefficient from assuming the resistance to be proportional to the momentum flux $\propto \rho_{\text{air}}U^2$ and using a pendulum to observe the progressive decrease in amplitude due to this drag force.

You can use the following expression, which will give you a rough, but possibly sufficiently accurate value for your purposes (which I ignore):

$$ F_{\text{damp}} = 1/2 C_D \rho_{\text{air}} A U(t)^2 $$ where A is the crossectional area of your mass, $U$ is the velocity and $C_D$ is the drag coefficient, which depends on the particular geometry of your mass. For tabulated values of $C_D$ you can look in the Wikipedia article. In any case, always bear in mind that this force is only an average, since for high Reynolds numbers, the flow of air around the mass will not be steady, but highly oscillatory due to turbulence.

Only For the case of very small Reynolds number do you have a damping that is steady and proportional to the velocity (and not the velocity squared). This is the Stokes drag, which for a sphere, can be calculated as

$$F_{\text{damp}} = 6\pi\rho_{\text{air}}\nu R U(t)$$

where $R$ is the radius of the sphere.

I hope this helps. Note that in all this, I have neglected effect of air on the spring itself, which is probably reasonable given the uncertainties involved.