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\author{Andrew Mark Allen - 05370299}
\title{Gravitational Torsion Pendulum}
\begin{document}
\maketitle
\section*{Abstract}
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The purpose of this experiment is to measure the gravitational constant G using a torsion pendulum. 
\section*{Introduction and Theory}
This experiment is fundamentally similar to that performed by Henry Cavendish in 1798, where a torsion balance is set up as in \todo{add diagram} and the graviational attraction between a pair of large balls and a pair of smaller balls is measured in terms of the torsion of the wire from which the smaller balls are suspended from. In our case, the torsion of the wire is measured by a method whereby a laser beam some distance ($0.67m$) away from the wire is reflected from a mirror attached to the wire onto a detector. \todo{add diagram}
\subsection*{Oscillation of the balance}
When the lead sphere is in position I \todo{add diagram}, it performs oscillations, in the plane of all the masses, about a particular equilibirum position $\alpha_1$. While in principle, the experiment can be left long enough for the oscillations to decay completely, in reality this isn't possible due to the eventuality of an outside force affecting the experiment. Any force affecting the motion of the masses will result in a change in the oscillation of the torsion balance. These forces will result in the vibration of the balance in several modes of oscillation, for instance a vertical up and down oscillation of the balance, or a pendulum like rocking of the balance in any direction. For this reason it was necessary to fit the relevant parts of the actual data to a general damped harmonic oscillator function, $y = Ae^{-\gamma t}\cos{(\omega t - \alpha)}$
\subsection*{Derivation of the formula for G}
In this experiment, aside from the various masses and distances between objects, as described below, the main measurements to be made were on the period of oscillation, $T$, of the torsion balance, and the points of equilibrium, $\alpha_1 and \alpha_2$, about which the balance oscillated. From these then, can the gravitational constant G be measured. The argument proceeds thus:
\begin{itemize}
\item
Let $b = \text{Distance between the centres of mass} = 0.047m$;
\item
$d = \text{Distance of the midpoints of the lead balls from the axis} = 50\text{mm}$
\item
$M = \text{Mass of one big lead sphere} = (1,500 \pm 5)g$;
\item
$L = \text{Distance between the mirror and the detector} = \text{0.67m}$
\end{itemize}
When the pendulum is in equilibrium, the torque due to the gravitational attraction between the spheres is equal to that due to the torsion of the string \[k\alpha_1 = \tau_1\]
where $\tau_1$ is the gravitational torque and k is the constant of torsion of the wire. Similarily we have the expression for $\alpha_2$, $k\alpha_2 = \tau_2$. Summing the two gives 

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\section*{Experimental Method}

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\section*{Results and Analysis}
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From the first graph, we have 
\section*{Discussion and Conclusion}
The general theme to the entire experiment was the difficulty in getting a set of usable results without the apparatus being disturbed. The entire run for the first week did not result in any usable results, while the results from the second week were only usable thanks to the help from Prof. Stamenov with using nonlinear regression to fit a curve to a subset of data which hadn't been contaminated (detected by looking for unexpected inflections in the curve).
 
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%COMMENT: HOW TO INCLUDE GRAPHICS
%\begin{figure}[h!]
%\centering
%\fbox{\includegraphics[width=0.9\textwidth]{Graph2.png}}
%\caption{Plot of $y^2$ against ring number}
%\label{fig:graph2}
%\end{figure}
%The experimental results for the travelling microscope experiments are shown in Figure \ref{fig:graph1} and Figure \ref{fig:graph2}