Propagation of errors

This project deals with a common situation in science: when measurements have some uncertainties.

For example, as a way to obtain the value of the acceleration of gravity, one can use a pendulum. I can measure the period (time to complete a cycle) of a pendulum of length \( L=1 {\rm m} \) in the lab 10 times, and obtain the results, in seconds: $$ T = 1.99, 2.05, 1.99, 2.00, 1.98, 2.00, 2.00, 2.01, 1.99, 2.00 $$

The question is: what value do I use as result of my experiment? How do I quantify the uncertainty of the measurement?

The answers are that one should use the mean of the values as the result of the experiment, and the standard error. You will need to learn what the previous sentences are, and how using this procedure one gets as final value of the experiment $$ T = 2.001 \pm 0.006 $$ which means that, given the data that we collected, the true value of the period lay between \( 2.001-0.006 = 1.995 {\rm sec} \) and \( 2.001+0.006 = 2.007 {\rm sec} \) (being precise, a 68% of the cases will be in the interval \([1.995,2.007]\), and a 95% of the cases in the interval \([1.989,2.014]\)).

Once we have our value for the period \( T = 2.001 \pm 0.006 \), we want to use it to compute the value of the acceleration of gravity. The relation between the period of a pendulum and the acceleration of gravity is given by the equation $$ g = \frac{4\pi^2 L}{T^2} $$

The problem that we have to solve now is how to “translate” the uncertainty on the period (\( T \)) in an uncertainty in \(g\). The question basically amounts to answer how much changes \(g\) if (\( T \)) is changed by \( \pm 0.006 \) around \( T = 2.001\). The answer is obtained by using the linear approximation studied in class, although it can be found in many places, like wikipedia, or sciencemastery.

In this project you will need to understand how to propagate uncertainties from one variable to another, and understand this in terms of linear approximations. You will also need to learn how to combine uncertainties in more than one variable. Ideally in this project you will provide many useful examples, similar to the one of the pendulum.