# How to Determine Viscosity using Damped Oscillations? [closed]

How does one find out the viscosity of a fluid, using viscous damping of an oscillating spring? Any help would be greatly appreciated.

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• Possible duplicate of How to calculate viscous damping coefficient? Oct 2, 2015 at 0:15

You have to make an additional assumption about how the friction force is affected by the viscosity. For small speeds you can assume Stokes flow around the oscillating mass. If the mass is a ball with radius $a$, then the friction force $F$ is given by velocity $v$ and dynamic viscosity $\mu$ by this relation: $$F = -6 \pi \mu a v$$
From the solution to the harmonic oscillator equation we know, that for the object on a spring damped by the force given by $F = -bv$ (so in our stokes flow, $b = 6 \pi \mu a$), the resulting motion $x(t)$ will obey this equation:
$$x(t) = A e^{-\frac{2b}{m}t} \sin(\omega t + \phi_0)$$
(this is valid for under-damped case, when the damping is small enough that the object can oscillate, generally see https://en.wikipedia.org/wiki/Harmonic_oscillator). In this equation $m$ is the mass of the object, $\omega$ is the natural frequency of the oscillator, determined by the $b$ and spring stiffness, and $A$ and $\phi_0$ are constants determined by initial conditions.
For us it means, that the amplitudes of the oscillation decrease with time exponentially as $e^{-\frac{2b}{m}t}$. Therefore, if we measure the times and amplitudes of several oscillations, we can determine the $\frac{2b}{m}$ constant an in turn, compute the viscosity $\mu$ using the equation $b = 6 \pi \mu a$.