Partnership for Integration of Computation into Undergraduate Physics

## 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!_