# Damped Harmonic Curve fit and Force

I am doing an experiment which includes looking at how the damping constant is affected when you attach circular disks with different surface area to a spring mass system. It is also submerged in a beaker filled with water. In order to obtain the best raw data I had to use a force sensor instead of a motion detector and now I am having trouble with my calculations.

(Damping is not part of my curriculum, so I have had to learn about it independently)

In Logger Pro I have the option to "Damped Harmonic" curve fit, but I have only seen those equations with amplitude vs time graphs, not force vs time. Is it possible to use this curve fit in order to find the damping coefficient?

If not, are there any calculations I can do with this data that can create a graph with the surface area of the disks on the x-axis and some other value on the y-axis?

Thank you :)

## 1 Answer

Let's take the equation of motion : $$m \dfrac{d^2x}{dt^2} + \gamma\dfrac{dx}{dt} + \omega_0^2 x = 0$$

The damping coefficient is thus $\gamma$. The solution for the motion of an under-damped oscillator follows (assuming $x=0$ at equilibrium) : $$x(t) = \lambda e^{-\gamma t/2m}\cos (\omega t+\phi)$$ The force is $F=m\dfrac{d^2x}{dt^2}$, so it is proportional to the second derivative of $x$. Derivating $x$ will not affect the argument of the exponential, and so this argument will still be $-\gamma t$ in each exponential in the resulting expression of $F$. As a result, the value of $B$ in your fit parametrization should be equal to $\gamma$. The same reasoning applies to the cosine function : the force will oscillate at the same frequency as the mass attached to the spring. The oscillation pulsation won't be exactly $\omega_0$, but rather : $$\omega = \omega_0\sqrt{1-\dfrac{\gamma^2}{4m^2\omega_0^2}}$$