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This law states that the magnitude of the gravitational force between two masses is - % Scalar (magnitude) form -\[ -F = G \frac{m_1 m_2}{r^2}, -\] +\begin{equation} +F = G \frac{m_1 m_2}{r^2} +\label{eq:NLUG} +\end{equation} where \(F\) is the gravitational force, \(m_1\) and \(m_2\) are the interacting masses, \(r\) is the distance between their centers. \(G\) is the universal gravitational constant, a constant of proprtionality that has been calculated to be @@ -180,58 +184,233 @@ Note that the above steps require the following raw data the be collected at eac \section*{Results} + + \subsection*{Raw Data} -\begin{table}[h!] +While all data was collected jointly, the five separate experimental setups can be split up into the following tables for convenience: + + + +\begin{table}[H] \centering -\caption{Measured critical angles for static and kinetic friction.} -\label{tab:friction_angles} -\begin{tabularx}{0.8\textwidth}{@{}lcc@{}} +\caption{Force between two masses while varying $m_1$.} +\label{tab:grav_m1} + +% Increase horizontal space between columns + +% Slightly increase vertical spacing +\renewcommand{\arraystretch}{1.3} + +\begin{tabularx}{0.95\textwidth}{ + @{} + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + @{} +} \toprule -\textbf{Trial} & \textbf{Static Friction Angle ($^\circ$)} & \textbf{Kinetic Friction Angle ($^\circ$)} \\ +\textbf{Trial} & +\textbf{$m_1$ (kg)} & +\textbf{$m_2$ (kg)} & +\textbf{$x_1$ (m)} & +\textbf{$x_2$ (m)} & +\textbf{$F_{1\rightarrow2}$ (N)} & +\textbf{$F_{2\rightarrow1}$ (N)} \\ \midrule -1 & 19.0 & 15.0 \\ -2 & 17.5 & 14.5 \\ -3 & 18.5 & 15.0 \\ -4 & 18.0 & 16.0 \\ -5 & 19.0 & 14.0 \\ -\midrule -\textbf{Average} & 18.4 & 14.9 \\ +1 & 50 & 100 & 2.00 & 6.00 & $2.09\times10^{-8}$ & $2.09\times10^{-8}$ \\ +2 & 100 & 100 & 2.00 & 6.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\ +3 & 250 & 100 & 2.00 & 6.00 & $1.04\times10^{-7}$ & $1.04\times10^{-7}$ \\ +4 & 500 & 100 & 2.00 & 6.00 & $2.09\times10^{-7}$ & $2.09\times10^{-7}$ \\ +5 & 750 & 100 & 2.00 & 6.00 & $3.13\times10^{-7}$ & $3.13\times10^{-7}$ \\ +6 & 1000 & 100 & 2.00 & 6.00 & $4.17\times10^{-7}$ & $4.17\times10^{-7}$ \\ \bottomrule \end{tabularx} \end{table} +\begin{table}[H] +\centering +\caption{Force between two masses while varying $m_2$.} +\label{tab:grav_m2} +\renewcommand{\arraystretch}{1.3} +\begin{tabularx}{0.95\textwidth}{ + @{} + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + @{} +} +\toprule +\textbf{Trial} & +\textbf{$m_1$ (kg)} & +\textbf{$m_2$ (kg)} & +\textbf{$x_1$ (m)} & +\textbf{$x_2$ (m)} & +\textbf{$F_{1\rightarrow2}$ (N)} & +\textbf{$F_{2\rightarrow1}$ (N)} \\ +\midrule +1 & 100 & 50 & 2.00 & 6.00 & $2.09\times10^{-8}$ & $2.09\times10^{-8}$ \\ +2 & 100 & 100 & 2.00 & 6.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\ +3 & 100 & 250 & 2.00 & 6.00 & $1.04\times10^{-7}$ & $1.04\times10^{-7}$ \\ +4 & 100 & 500 & 2.00 & 6.00 & $2.09\times10^{-7}$ & $2.09\times10^{-7}$ \\ +5 & 100 & 750 & 2.00 & 6.00 & $3.13\times10^{-7}$ & $3.13\times10^{-7}$ \\ +6 & 100 & 1000 & 2.00 & 6.00 & $4.17\times10^{-7}$ & $4.17\times10^{-7}$ \\ +\bottomrule +\end{tabularx} +\end{table} -The coefficients of static and kinetic friction were calculated using the relationship: -\[ -\mu = \tan(\theta) -\] -where $\theta$ is the average critical angle measured for each case. +\begin{table}[H] +\centering +\caption{Force between equal masses while varying $m_1 = m_2$.} +\label{tab:grav_equal} +\renewcommand{\arraystretch}{1.3} +\begin{tabularx}{0.95\textwidth}{ + @{} + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + @{} +} +\toprule +\textbf{Trial} & +\textbf{$m_1$ (kg)} & +\textbf{$m_2$ (kg)} & +\textbf{$x_1$ (m)} & +\textbf{$x_2$ (m)} & +\textbf{$F_{1\rightarrow2}$ (N)} & +\textbf{$F_{2\rightarrow1}$ (N)} \\ +\midrule +1 & 50 & 50 & 2.00 & 6.00 & $1.04\times10^{-8}$ & $1.04\times10^{-8}$ \\ +2 & 100 & 100 & 2.00 & 6.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\ +3 & 250 & 250 & 2.00 & 6.00 & $2.61\times10^{-7}$ & $2.61\times10^{-7}$ \\ +4 & 500 & 500 & 2.00 & 6.00 & $1.04\times10^{-6}$ & $1.04\times10^{-6}$ \\ +5 & 750 & 750 & 2.00 & 6.00 & $2.35\times10^{-6}$ & $2.35\times10^{-6}$ \\ +6 & 1000 & 1000 & 2.00 & 6.00 & $4.17\times10^{-6}$ & $4.17\times10^{-6}$ \\ +\bottomrule +\end{tabularx} +\end{table} -Average angle for static friction: -\[ -\theta_s = 18.4^\circ -\] +\begin{table}[H] +\centering +\caption{Force between two masses while varying $x_1$.} +\label{tab:grav_dist1} +\renewcommand{\arraystretch}{1.3} -Coefficient of static friction: -\[ -\mu_s = \tan(18.4^\circ) = 0.33 -\] +\begin{tabularx}{0.95\textwidth}{ + @{} + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + @{} +} +\toprule +\textbf{Trial} & +\textbf{$m_1$ (kg)} & +\textbf{$m_2$ (kg)} & +\textbf{$x_1$ (m)} & +\textbf{$x_2$ (m)} & +\textbf{$F_{1\rightarrow2}$ (N)} & +\textbf{$F_{2\rightarrow1}$ (N)} \\ +\midrule +1 & 100 & 100 & 0.00 & 10.00 & $6.67\times10^{-9}$ & $6.67\times10^{-9}$ \\ +2 & 100 & 100 & 2.00 & 10.00 & $1.04\times10^{-8}$ & $1.04\times10^{-8}$ \\ +3 & 100 & 100 & 4.00 & 10.00 & $1.85\times10^{-8}$ & $1.85\times10^{-8}$ \\ +4 & 100 & 100 & 6.00 & 10.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\ +5 & 100 & 100 & 8.00 & 10.00 & $1.67\times10^{-7}$ & $1.67\times10^{-7}$ \\ +\bottomrule +\end{tabularx} +\end{table} +\begin{table}[H] +\centering +\caption{Force between two masses while varying $x_2$.} +\label{tab:grav_dist2} +\renewcommand{\arraystretch}{1.3} -Average angle for kinetic friction: -\[ -\theta_k = 14.9^\circ -\] +\begin{tabularx}{0.95\textwidth}{ + @{} + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + >{\centering\arraybackslash}X + @{} +} +\toprule +\textbf{Trial} & +\textbf{$m_1$ (kg)} & +\textbf{$m_2$ (kg)} & +\textbf{$x_1$ (m)} & +\textbf{$x_2$ (m)} & +\textbf{$F_{1\rightarrow2}$ (N)} & +\textbf{$F_{2\rightarrow1}$ (N)} \\ +\midrule +1 & 100 & 100 & 0.00 & 10.00 & $6.67\times10^{-9}$ & $6.67\times10^{-9}$ \\ +2 & 100 & 100 & 0.00 & 8.00 & $1.04\times10^{-8}$ & $1.04\times10^{-8}$ \\ +3 & 100 & 100 & 0.00 & 6.00 & $1.85\times10^{-8}$ & $1.85\times10^{-8}$ \\ +4 & 100 & 100 & 0.00 & 4.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\ +5 & 100 & 100 & 0.00 & 2.00 & $1.67\times10^{-7}$ & $1.67\times10^{-7}$ \\ +\bottomrule +\end{tabularx} +\end{table} -Coefficient of kinetic friction: -\[ -\mu_k = \tan(14.9^\circ) = 0.27 -\] +\subsection*{Analysis} + +The first major observation that can be made using the raw data is that for all data points (in all 5 tables), +\textbf{$F_{1\rightarrow2}$ (N)} = \textbf{$F_{2\rightarrow1}$ (N)}. This observation is \textbf{Newton's Third Law}, +as the force of one mass on another is equal to the other mass on it. Hence, these two forces can be replaced by one +force column, denoted as $F_g$, which is the force of gravitational attraction between the two objects. Also notes that +the direction of these two forces are always towards each others, which is why gravitation is known as a force of attraction. \\\\ -Note that the minute acceleration due to the tapping of the block is neglected, as it's acceleration is assumed to be negligible. Also neglected is the force of air resistance, as it is assumed to be very small compared to the other forces acting on the block. +Then, look to Table \ref{tab:grav_m1}. Observe that there is a relationship between the changing variable ($m_1$) +and the gravitational force. As the other variables are all constant in this experiment, the relationship between +$m_1$ and $F_g$ can be graphed as follows. + + + +\begin{figure}[h!] % h! = “here” placement + \centering + \includegraphics[width=0.7\textwidth]{Force vs m1} % <-- your image file name + \caption{$F_g$ vs $m_1$ graphed} + \label{fig:m1graph} +\end{figure} + + +Observing Figure \ref{fig:m1graph}, there is a linear trend between the gravitational force and the mass of object 1. +This can be represented by the proportionality: +\[ +F_g \propto m_1 +\] +Hence, the relationship can be described witht eh below equation, where $k_1$ is simply a constant: + +\begin{equation} +F_g = k_1 \times m_1 +\label{eq:grav_force} +\end{equation} + +Applying the same logic to Table \ref{tab:grav_m2} yields the following relationship between $F_g$ and $m_2$ + + \section*{Error Analysis}