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= “here” placement +\begin{figure}[H] % h! = “here” placement \centering \includegraphics[width=0.7\textwidth]{Force vs Mass m1 m2} % <-- your image file name \caption{$F_g$ vs mass of $m_1$ = $m_2$} @@ -458,8 +458,90 @@ Looking at Table \ref{tab:grav_equal}, this derived relationship can be verified As suggested by equation \ref{eq:m1and2}, there is a proportional quadratic relationship between the masses of the objects and the resulting gravitational force (see Figure \ref{fig:m1m2graph}). -Therefore, equation \ref{eq:m1and2} is validated by the simulation. +Therefore, equation \ref{eq:m1and2} is validated by the simulation.\\\\ +Moving onto Tables \ref{tab:grav_dist1} and \ref{tab:grav_dist2}, the output columns for the gravitational force appear identical. +Upon further inspection, the gravitational force seems to exhibit a dependency on the distance between the masses. +A processed data table can be made combining Tables \ref{tab:grav_dist1} and \ref{tab:grav_dist2}, +showcasing the relationship between $F_g$ and $r:= |x_1-x_2|$. + +\begin{table}[H] +\centering +\caption{Gravitational force versus distance between two masses.} +\label{tab:grav_vs_r} +\renewcommand{\arraystretch}{1.3} + +\begin{tabularx}{0.8\textwidth}{ + @{} + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + @{} +} +\toprule +\textbf{Distance $r$ (m)} & \textbf{Gravitational Force $F_g$ (N)} \\ +\midrule +2.00 & $1.67\times10^{-7}$ \\ +4.00 & $4.17\times10^{-8}$ \\ +6.00 & $1.85\times10^{-8}$ \\ +8.00 & $1.04\times10^{-8}$ \\ +10.00 & $6.67\times10^{-9}$ \\ +\bottomrule +\end{tabularx} +\end{table} + +This new relationship is graphed in the figure below: + + +\begin{figure}[H] % h! = “here” placement + \centering + \includegraphics[width=0.7\textwidth]{Force vs Distance Between Masses} % <-- your image file name + \caption{$F_g$ vs $r= x_2 - x_1$} + \label{fig:rgraph} +\end{figure} + +The shown trendline suggests a proprtional fit to $\frac{1}{r^2}$ as follows: + +\[ +F_g \propto \frac{1}{r^2}, +\] + +so + +\begin{equation} +F_g = k_4 \times \frac{1}{r^2}. +\label{eq:finalEq} +\end{equation} + +When combined with equation \ref{eq:m1and2}, NLUG pops out, and the proprtionality constant can be denotes as $G$: + + +\begin{equation} +F_g = G \frac{m_1 m_2}{r^2} \tag{\ref{eq:NLUG}} +\end{equation} + +The final step is to solve for the proportionality constant, $G$. As the relationship has been proven, any data point can be used to solve for this constant. +For simplicity, the control data point (used for setup) will be used, where $m_1$ = 100 kg, $m_2$ = 100 kg, $r = x_2 - x_1 = 6\,\text{m} - 2\,\text{m} = 4\,\text{m}$. +We have +\[ +F_g = G \left( \frac{m_1 m_2}{r^2} \right) += G \left( \frac{100\,\text{kg} \times 100\,\text{kg}}{4^2\,\text{m}^2} \right) += 4.17 \times 10^{-8}\,\text{N} +\] + +\[ +G = \frac{F_g \, r^2}{m_1 m_2} += \frac{(4.17 \times 10^{-8}\,\text{N}) (4\,\text{m})^2}{100\,\text{kg} \times 100\,\text{kg}} += 6.67 \times 10^{-11}\,\text{N\,m}^2\text{/kg}^2 +\]. + +Ergo, the data from the simulation can be used to derive NLUG and solve for $G$, the universal gravitational constant: + + +\begin{equation} +F_g = G \frac{m_1 m_2}{r^2}, \quad +G = 6.67 \times 10^{-11}\,\text{N\,m}^2\text{/kg}^2 +\label{eq:solvedNLUG} +\end{equation} \section*{Error Analysis}