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Curve Fitting
Developed by Eric Ayars - Published August 1, 2016
Being able to fit a model to experimental data is an extremely important laboratory skill. Most physics students are familiar with linear curve fitting, often with a spreadsheet or data-collection software such as _Data Studio_ (PASCO Scientific) or _Logger Pro_ (Vernier Software).
This set of exercises takes students beyond these introductory tools into the realm of arbitrary-function curve fitting, with error bars and estimates of parameter uncertainties.
| Subject Areas | Mathematical/Numerical Methods and Experimental Labs |
|---|---|
| Level | Beyond the First Year |
| Available Implementation | Python |
| Available Variation |
Non Linear Curve Fitting |
| Learning Objectives |
Students who complete this module will gain experience in
* Reading data files into the computer.
* Fitting data to arbitrary functions.
* Using uncertainty in data points as part of the calculation of the curve fit.
* Reporting parameter values _with uncertainty_.
|
| Time to Complete | 90 min |
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
## Exercise 1: Linear fit
The data set "calibration.txt" shows the _reported_ position of a rotational sensor (in units of $\frac{1}{1024}$ of a rotation) after $N$ revolutions. If the rotational sensor is properly calibrated, this should be a horizontal line at 0, but it's not.
* By how much per revolution is the sensor miscalibrated? Plot a graph of the data, with a linear curve fit, to answer. Be sure to include errorbars on the graph, and report the uncertainty in your fit parameters.
## Exercise 2: Power fit
Students in a first-semester physics course collected this data showing the angular velocity of a rotating "point mass" $m$ and the force $F_c$ required to keep the mass rotating in a circle of radius $R$. If you plot this data, it _looks_ linear (except for that one point at (0,0)) but for theoretical reasons we believe that the equation for centripetal force should be
$$ F_c = mR\omega^2 $$
* Does this data fit the model for $F_c$? Plot a graph to support your answer.
* Statistically, the errorbars should intersect the curve fit for about 63% of the data points: Are your errorbars reasonable?
* The rotating mass had $m=200$ grams, and the radius of rotation was $R=18$ cm. Is this consistent with parameters from your curve fit?
## Exercise 3: Exponential fit
Barium 137 is radioactive. Activity of a sample of Barium 137 was measured as a function of time, and the results are shown in file 'decay.txt'. We would expect that for radioactive decay,
$$ N = N_o e^{-t/\tau} $$
where $\tau= \frac{t_{1/2}}{\ln(2)}$.
* Find the half-life $t_{1/2}$ for Barium 137. Support your answer with a graph and a curve fit, of course!
## Exercise 4: Challenge!
The data in file damped_oscillation.txt is shows the position of a magnetic rotor in a fixed magnetic field. One model for the behavior of this rotor would be "exponentially-damped oscillation",
$$ \theta = \theta_o e^{-\beta t} \cos(\omega t + \phi) + \varphi $$
Find the resonant frequency $\omega$ and the damping constant $\beta$ for this apparatus. You may assume the uncertainty in position is negligible.
_Hint: Your initial guesses need to be in the right ballpark!_
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Credits and Licensing
Eric Ayars, "Curve Fitting," Published in the PICUP Collection, August 2016.
The instructor materials are ©2016 Eric Ayars.
The exercises are released under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license
