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\title{The Scientific Study of The Gravitational Force of The Earth When Acting on a Body in Simple Harmonic Motion}
\author{Tommy Cunningham}
\date{October 2024}



\begin{document}
\maketitle
\section{Introduction}
The concept of uniform acceleration due to gravity has been studied since the 17th century, most notably by Galileo. His famous (but likely mythical) Leaning Tower of Pisa experiment is said to have demonstrated that objects of different masses fall at the same rate when air resistance is negligible. This principle laid the foundation for the study of gravitational acceleration  . Simple harmonic motion is believed to have also been discovered by Galileo, however wasn't expanded on until the 19th century when John Fourier demonstrated that any object that completes a repeating motion no matter how complex can be analysed as sum of simple harmonic motions.
We can use both of these concepts on a simple pendulum to work out the acceleration caused by the gravitational force of the earth on an object with constantly changing velocity, ie. a simple pendulum. In doing so we can test to see if acceleration is uniform on a body. And we can also prove that:  \(L  \propto  T^2\)


\section{Method}
Using a string and a nut (which acts as our mass). Using a metre rule we cut the string down to a length of \(60cm\), this allowed for room to tie a knot around the mass. We would take our first measurement of \(L\) form \(55cm\) . We would take measurements in integers of 10, down to 5cm, taking a reading for how long it takes the mass to oscillate 10 times (this allows us to have a much smaller uncertainty for 1 oscillation). We would also repeat the experiment 3 times to allow us to take an average which lowers the random error.




  
    \begin{figure}
    \section{Table}
        \includegraphics[width=1.2\linewidth]{SmartSelect_20241003_110314_Microsoft 365 (Office).jpg}
        \caption{Table of measurements for different lengths and their corresponding times.}
        \label{fig:enter-label}
    \end{figure}
    
  
    \begin{figure}
        \section{Graph}       
        \centering
        \includegraphics[width=1.3\linewidth]{SmartSelect_20241003_123225_Microsoft 365 (Office).jpg}
        \caption{Graph plotted using values from the table with L as a function of \(T^2\). Along with the max. and min. gradient lines.}
        \label{fig:enter-label}
    \end{figure}

    
    
    


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