Physics Lab Report

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\sisetup{separate-uncertainty=true}

\title{Lab 4: Measurement of $g$ Using a Simple Pendulum}
\author{First Last \\ Partner: Jane Doe \\ TA: Dr. Example}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We measure the acceleration due to gravity by timing the period of a simple
pendulum over a range of lengths. Our result, $g = \SI{9.78(3)}{\meter\per\second\squared}$,
agrees with the accepted value \SI{9.81}{\meter\per\second\squared} within experimental uncertainty.
\end{abstract}

\section{Introduction}
For small oscillations, a simple pendulum of length $L$ has period
\begin{equation}
  T = 2\pi\sqrt{L/g}.
  \label{eq:pendulum}
\end{equation}
A linear fit of $T^2$ versus $L$ has slope $4\pi^2/g$.

\section{Apparatus}
\begin{itemize}
  \item Light string ($\le\SI{1}{\gram}$), stopwatch (\SI{\pm 0.01}{\second}).
  \item Steel bob, \SI{100}{\gram}; meter stick (\SI{\pm 0.1}{\centi\meter}).
  \item Ring stand, clamp.
\end{itemize}

\section{Procedure}
We varied $L$ from \SI{20}{\centi\meter} to \SI{100}{\centi\meter} in \SI{10}{\centi\meter} steps.
For each length we displaced the bob about \ang{5} and timed 10 oscillations,
repeating three times.

\section{Data}
\begin{table}[H]
  \centering
  \caption{Mean period $T$ vs. length $L$.}
  \begin{tabular}{ccc}
    \toprule
    $L$ (\si{\centi\meter}) & $T$ (\si{\second}) & $T^2$ (\si{\second\squared}) \\
    \midrule
    20.0 & 0.898 & 0.806 \\
    40.0 & 1.270 & 1.613 \\
    60.0 & 1.557 & 2.424 \\
    80.0 & 1.797 & 3.229 \\
   100.0 & 2.009 & 4.036 \\
    \bottomrule
  \end{tabular}
\end{table}

\section{Analysis}
A linear regression gives slope $m = \SI{4.04(3)}{\second\squared\per\meter}$.
From $m = 4\pi^2/g$ we find
\[
  g = \frac{4\pi^2}{m} = \SI{9.78(3)}{\meter\per\second\squared}.
\]
Uncertainty propagation: $\delta g / g = \delta m / m$.

\section{Discussion}
The small-angle approximation introduces a systematic bias of $\sim 0.05\%$.
The dominant uncertainty is human timing. Agreement with the accepted value is
within $\sim 1\sigma$.

\section{Conclusion}
A simple pendulum is a practical and accurate instrument for measuring $g$.

\end{document}