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# Lecture Notes

Below are the lecture notes from my course on multivariable calculus, as well as some handouts I give to students for some of the experiments in the introductory physics lab courses. Note that I update my notes each term, but this website may not link to the latest editions.

### Notes on IPL/CPS Physics Labs

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Notes on Experiment 12: The Simple Pendulum

Sujeet Akula

Error Analysis

Varying the Length

Here we have a set of data $\{\ell_j,m_j,\{t_{i,j}\}\}_{j=1}^{j=3}$, though $m$ is fixed for every $j$, where $j$ refers to different configuration of the pendulum length, and $i=1,2,3,4,5$ for each time trial. \begin{align} \end{align}First, we averate $\{t_{i,j}\}$ for each $j$, and also find the standard deviation\begin{align} t_j = \ave\left\{t_{i,j}\right\} &\text{ and } \sigma_{t_j} = \std\left\{t_{i,j}\right\} \end{align}This gives the overall uncertainty in $t$ to be\begin{align} \delta t_j &= \sqrt{\sigma_{t_j}^2 + \Delta_t^2} , \end{align}where $\Delta_t = 0.1\,\mathrm{s}$ is the systematic uncertainty. Now we can compute the period:\begin{align} T_j &= \dfrac{1}{10}t_j , \end{align}and the uncertainty\begin{align} \delta T_j &= \dfrac{1}{10}\delta t_j . \end{align}Of course, we wish to plot $T^2$, which has uncertainty\begin{align} \delta(T_j^2) &= 2 T_j \delta T_j . \end{align} We should now have the set $\{\ell_j,m_j,T_j^2\}_{j=1}^{j=3}$, with uncertainties in $T^2$, $\{\delta(T_j^2)\}_{j=1}^{j=3}$. Plot $\ell$ on the vertical axis and $T^2$ on the horizontal axis, where $\delta(T^2)$ are the horizontal error bars. Use http://freeboson.org/slope to compute the slope, $\mu$ and uncertainty in slope, $\delta \mu$. \begin{align} \end{align}Knowing that\begin{align} \omega &= \sqrt{\dfrac{g}{\ell}} , \end{align}and\begin{align} \omega &= \dfrac{2\pi}{T} , \end{align}we can write that\begin{align} \ell &= \dfrac{g}{4\pi^2} T^2 . \end{align}This means that\begin{align} g = 4\pi^2\mu &\text{ and } \delta g = 4\pi^2 \delta\mu , \end{align} our final results.