613 lines
24 KiB
TeX
613 lines
24 KiB
TeX
\documentclass[12pt,letterpaper]{article}
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\begin{document}
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% ---------- SIMPLE HEADER ----------
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\pagenumbering{arabic} % start with Arabic page numbers
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\setcounter{page}{1}
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% Header: title centered, partner left, date right
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\begin{center}
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\textbf{\large Experimentally Deriving Newton’s Universal Law of Gravitation by Measuring Mass, Distance, and G in a Simulation}
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\end{center}
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\vspace{1em}
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\noindent
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\textbf{Name:} Keshav Anand
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\hfill
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\textbf{Date:} 16 November 2025
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% ---------- MAIN DOCUMENT ----------
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\section*{Objectives}
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\begin {itemize}
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\item Use the \href{http://phet.colorado.edu/sims/html/gravity-force-lab/latest/gravity-force-lab_en.html}{Gravitational force simulation} to determine the dependence of the gravitational force on the mass of the objects involved.
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\item Use the same simulation to determine the dependence of the gravitational force on the distance between the two masses.
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\item Determine the experimental value of the universal gravitational constant (G). (This is what relates the gravitational force to the masses and distance rather than being these proportional. G must be included in your final equation.)
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\item Determine an Equation for the Universal Law of Gravitation based on your data, using only symbols.
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\end {itemize}
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\section*{Introduction}
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\subsection*{Background}
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The concept of gravitational force has been misunderstood for most of history,
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commonly attributed to divine or supernatural causes until Sir Isaac Newton published his groundbreaking \textit{Principia}.
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In the Principia, Newton asserted that every mass exerts an attractive force on every other mass,
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a phenomenon described by Newton's Universal Law of Gravitation (NLUG).
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This law states that the magnitude of the gravitational force between two masses is
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\begin{equation}
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F_g = G \frac{m_1 m_2}{r^2}
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\label{eq:NLUG}
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\end{equation}
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where \(F\) is the gravitational force, \(m_1\) and \(m_2\) are the interacting masses, \(r\) is the distance between their centers.
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\(G\) is the universal gravitational constant, a constant of proportionality that has been calculated to be
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\[
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G = 6.674 \times 10^{-11}\ \text{N}\,\text{m}^2\,\text{kg}^{-2}.
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\]
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In the scientific community, NLUG is treated as an absolute truth, and many important discoveries and applications
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rely on its accuracy. From engineering to astrophysics, NLUG has profound importance, and its validity
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is vital to the functioning of scientific advancement.
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\subsection*{Purpose}
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This lab aims to use a computer simulation
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to verify NLUG by deriving the relationship between gravitational force, masses of objects, and the distance
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between them. First, a gravitation simulation will be used to derive the relationship between objects'
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masses and the gravitational force. Then, the same process will be repeated with comparing gravitational
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force to the distance between objects. Data collected from these two setups will be used to (hopefully) re-establish
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the relationship of proportionality proposed by Newton, and the collected data will also solve for the
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universal Gravitational constant, $G$.
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\subsection*{Research Problem}
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The research objective for this project is to verify the gravitational relationship between two objects and verify the
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constant $G$. The primary problem of investigation is that theoretical mathematics often fails to adequately capture
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a true relationship in the real world. Moreover, using physical objects and tools of measurements can result in
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unwanted noisy data and is limited by the precision of measurement. Ergo, a simulation bridges this gap,
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allowing for an accurate verification of NLUG.
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\section*{Methodology}
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\subsection*{Materials and Resources}
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As this lab was performed within a simulation, all physical materials are limited to a computer with at least 400 MB of memory to render the simulation.
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Within the simulation, the simulated materials include
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\begin{itemize}
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\item Adjustable Mass, $m_1$
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\item Adjustable Mass, $m_2$
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\item 10 Meter Scale
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\item Automatic Force Scale to Measure Gravitational Attraction
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\item Two simulated people holding $m_1$ and $m_2$ from colliding into each other due to gravitation
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\end{itemize}
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\subsection*{Experimental Setup}
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\begin{figure}[h!] % h! = “here” placement
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\centering
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\includegraphics[width=0.7\textwidth]{ExperimentalSetup} % <-- your image file name
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\caption{Experimental setup for the gravity simulation}
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\label{fig:friction_setup}
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\end{figure}
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Note that in Figure \ref{fig:friction_setup}, all inputs (independent variables) are denoted in blue, whereas outputs (dependent variables) are denoted in red
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\subsection*{Procedure}
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\begin{enumerate}
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\item Set the location of mass 1 to exactly 2 meters on the scale, and set mass 2 to exactly 6 meters on the scale, with a distance between of 4 meters
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\item Set the mass of objects 1 and 2 to exactly 100 kg
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\item Set the force values to scientific notation, and uncheck the option for masses of constant size
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\item Leaving the mass of $m_2$ constant, change the mass of $m_1$ to be the values listed below, and record both the force on $m_1$ by $m_2$ and the force on $m_2$ by $m_1$
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\subitem Mass values for $m_1$ (kg): 50, 100, 250, 500, 750, 1000
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\item Reset the simulation as detailed by steps 1-3
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\item Leaving the mass of $m_1$ constant, change the mass of $m_2$ to be the values listed below, and record both the force on $m_1$ by $m_2$ and the force on $m_2$ by $m_1$
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\subitem Mass values for $m_2$ (kg): 50, 100, 250, 500, 750, 1000
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\item Reset the simulation as detailed by steps 1-3
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\item Change the masses of both $m_1$ and $m_2$ to be the values listed below, and record both the force on $m_1$ by $m_2$ and the force on $m_2$ by $m_1$
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\subitem Mass values for $m_1$ and $m_2$ (kg): 50, 100, 250, 500, 750, 1000
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\item Reset the simulation as detailed by steps 1-3
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\item Leave $m_2$ at 10 meters on the scale (align the black dot for center of mass), and move $m_1$ based on its center to the below values on the scale, and record both the force on $m_1$ by $m_2$ and the force on $m_2$ by $m_1$
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\subitem Position values for $m_1$ (m): 0, 2, 4, 6, 8
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\item Reset the simulation as detailed by steps 1-3
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\item Leave $m_1$ at 0 meters on the scale (align the black dot for center of mass), and move $m_2$ based on its center to the below values on the scale, and record both the force on $m_1$ by $m_2$ and the force on $m_2$ by $m_1$
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\subitem Position values for $m_1$ (m): 10, 8, 6, 4, 2
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\end{enumerate}
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Note that the above steps require the following raw data the be collected at each datapoint
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\begin{itemize}
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\item Position of the center of mass of $m_1$, (m)
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\item Position of the center of mass of $m_2$, (m)
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\item Mass of $m_1$, (kg)
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\item Mass of $m_2$, (kg)
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\item Force on $m_1$ by $m_2$, (N)
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\item Force on $m_2$ by $m_1$, (N)
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\end{itemize}
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\section*{Results}
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\subsection*{Raw Data}
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While all data was collected jointly, the five separate experimental setups can be split up into the following tables for convenience:
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\begin{table}[H]
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\centering
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\caption{Force between two masses while varying $m_1$.}
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\label{tab:grav_m1}
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% Increase horizontal space between columns
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% Slightly increase vertical spacing
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\renewcommand{\arraystretch}{1.3}
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\begin{tabularx}{0.95\textwidth}{
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@{}
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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@{}
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}
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\toprule
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\textbf{Trial} &
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\textbf{$m_1$ (kg)} &
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\textbf{$m_2$ (kg)} &
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\textbf{$x_1$ (m)} &
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\textbf{$x_2$ (m)} &
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\textbf{$F_{1\rightarrow2}$ (N)} &
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\textbf{$F_{2\rightarrow1}$ (N)} \\
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\midrule
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1 & 50 & 100 & 2.00 & 6.00 & $2.09\times10^{-8}$ & $2.09\times10^{-8}$ \\
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2 & 100 & 100 & 2.00 & 6.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\
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3 & 250 & 100 & 2.00 & 6.00 & $1.04\times10^{-7}$ & $1.04\times10^{-7}$ \\
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4 & 500 & 100 & 2.00 & 6.00 & $2.09\times10^{-7}$ & $2.09\times10^{-7}$ \\
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5 & 750 & 100 & 2.00 & 6.00 & $3.13\times10^{-7}$ & $3.13\times10^{-7}$ \\
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6 & 1000 & 100 & 2.00 & 6.00 & $4.17\times10^{-7}$ & $4.17\times10^{-7}$ \\
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\bottomrule
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\end{tabularx}
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\end{table}
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\begin{table}[H]
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\centering
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\caption{Force between two masses while varying $m_2$.}
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\label{tab:grav_m2}
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\renewcommand{\arraystretch}{1.3}
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\begin{tabularx}{0.95\textwidth}{
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@{}
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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@{}
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}
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\toprule
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\textbf{Trial} &
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\textbf{$m_1$ (kg)} &
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\textbf{$m_2$ (kg)} &
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\textbf{$x_1$ (m)} &
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\textbf{$x_2$ (m)} &
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\textbf{$F_{1\rightarrow2}$ (N)} &
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\textbf{$F_{2\rightarrow1}$ (N)} \\
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\midrule
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1 & 100 & 50 & 2.00 & 6.00 & $2.09\times10^{-8}$ & $2.09\times10^{-8}$ \\
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2 & 100 & 100 & 2.00 & 6.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\
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3 & 100 & 250 & 2.00 & 6.00 & $1.04\times10^{-7}$ & $1.04\times10^{-7}$ \\
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4 & 100 & 500 & 2.00 & 6.00 & $2.09\times10^{-7}$ & $2.09\times10^{-7}$ \\
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5 & 100 & 750 & 2.00 & 6.00 & $3.13\times10^{-7}$ & $3.13\times10^{-7}$ \\
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6 & 100 & 1000 & 2.00 & 6.00 & $4.17\times10^{-7}$ & $4.17\times10^{-7}$ \\
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\bottomrule
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\end{tabularx}
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\end{table}
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\begin{table}[H]
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\centering
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\caption{Force between equal masses while varying $m_1 = m_2$.}
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\label{tab:grav_equal}
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\renewcommand{\arraystretch}{1.3}
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\begin{tabularx}{0.95\textwidth}{
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@{}
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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@{}
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}
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\toprule
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\textbf{Trial} &
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\textbf{$m_1$ (kg)} &
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\textbf{$m_2$ (kg)} &
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\textbf{$x_1$ (m)} &
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\textbf{$x_2$ (m)} &
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\textbf{$F_{1\rightarrow2}$ (N)} &
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\textbf{$F_{2\rightarrow1}$ (N)} \\
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\midrule
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1 & 50 & 50 & 2.00 & 6.00 & $1.04\times10^{-8}$ & $1.04\times10^{-8}$ \\
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2 & 100 & 100 & 2.00 & 6.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\
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3 & 250 & 250 & 2.00 & 6.00 & $2.61\times10^{-7}$ & $2.61\times10^{-7}$ \\
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4 & 500 & 500 & 2.00 & 6.00 & $1.04\times10^{-6}$ & $1.04\times10^{-6}$ \\
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5 & 750 & 750 & 2.00 & 6.00 & $2.35\times10^{-6}$ & $2.35\times10^{-6}$ \\
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6 & 1000 & 1000 & 2.00 & 6.00 & $4.17\times10^{-6}$ & $4.17\times10^{-6}$ \\
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\bottomrule
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\end{tabularx}
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\end{table}
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\begin{table}[H]
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\centering
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\caption{Force between two masses while varying $x_1$.}
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\label{tab:grav_dist1}
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\renewcommand{\arraystretch}{1.3}
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\begin{tabularx}{0.95\textwidth}{
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@{}
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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@{}
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}
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\toprule
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\textbf{Trial} &
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\textbf{$m_1$ (kg)} &
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\textbf{$m_2$ (kg)} &
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\textbf{$x_1$ (m)} &
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\textbf{$x_2$ (m)} &
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\textbf{$F_{1\rightarrow2}$ (N)} &
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\textbf{$F_{2\rightarrow1}$ (N)} \\
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\midrule
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1 & 100 & 100 & 0.00 & 10.00 & $6.67\times10^{-9}$ & $6.67\times10^{-9}$ \\
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2 & 100 & 100 & 2.00 & 10.00 & $1.04\times10^{-8}$ & $1.04\times10^{-8}$ \\
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3 & 100 & 100 & 4.00 & 10.00 & $1.85\times10^{-8}$ & $1.85\times10^{-8}$ \\
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4 & 100 & 100 & 6.00 & 10.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\
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5 & 100 & 100 & 8.00 & 10.00 & $1.67\times10^{-7}$ & $1.67\times10^{-7}$ \\
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\bottomrule
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\end{tabularx}
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\end{table}
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\begin{table}[H]
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\centering
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\caption{Force between two masses while varying $x_2$.}
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\label{tab:grav_dist2}
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\renewcommand{\arraystretch}{1.3}
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\begin{tabularx}{0.95\textwidth}{
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@{}
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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>{\centering\arraybackslash}X
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@{}
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}
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\toprule
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\textbf{Trial} &
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\textbf{$m_1$ (kg)} &
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\textbf{$m_2$ (kg)} &
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\textbf{$x_1$ (m)} &
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\textbf{$x_2$ (m)} &
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\textbf{$F_{1\rightarrow2}$ (N)} &
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\textbf{$F_{2\rightarrow1}$ (N)} \\
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\midrule
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1 & 100 & 100 & 0.00 & 10.00 & $6.67\times10^{-9}$ & $6.67\times10^{-9}$ \\
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2 & 100 & 100 & 0.00 & 8.00 & $1.04\times10^{-8}$ & $1.04\times10^{-8}$ \\
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3 & 100 & 100 & 0.00 & 6.00 & $1.85\times10^{-8}$ & $1.85\times10^{-8}$ \\
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4 & 100 & 100 & 0.00 & 4.00 & $4.17\times10^{-8}$ & $4.17\times10^{-8}$ \\
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5 & 100 & 100 & 0.00 & 2.00 & $1.67\times10^{-7}$ & $1.67\times10^{-7}$ \\
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\bottomrule
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\end{tabularx}
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\end{table}
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\subsection*{Discussion and Analysis}
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The first major observation that can be made using the raw data is that for all data points (in all 5 tables),
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\textbf{$F_{1\rightarrow2}$ (N)} = \textbf{$F_{2\rightarrow1}$ (N)}. This observation is \textbf{Newton's Third Law},
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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
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force column, denoted as $F_g$, which is the force of gravitational attraction between the two objects. Also notes that
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the direction of these two forces are always towards each others, which is why gravitation is known as a force of attraction.
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\\\\
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Then, look to Table \ref{tab:grav_m1}. Observe that there is a relationship between the changing variable ($m_1$)
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and the gravitational force. As the other variables are all constant in this experiment, the relationship between
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$m_1$ and $F_g$ can be graphed as follows.
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\begin{figure}[h!] % h! = “here” placement
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\centering
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\includegraphics[width=0.7\textwidth]{Force vs m1} % <-- your image file name
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\caption{$F_g$ vs $m_1$ graphed}
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\label{fig:m1graph}
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\end{figure}
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Observing Figure \ref{fig:m1graph}, there is a linear trend between the gravitational force and the mass of object 1.
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This can be represented by the proportionality:
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\[
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F_g \propto m_1.
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\]
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Hence, the relationship can be described with the below equation, where $k_1$ is simply a constant:
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\begin{equation}
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F_g = k_1 \times m_1
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\label{eq:m1}
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\end{equation}
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Applying the same logic to Table \ref{tab:grav_m2} yields the following relationship between $F_g$ and $m_2$
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\begin{figure}[h!] % h! = “here” placement
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\centering
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\includegraphics[width=0.7\textwidth]{Force vs m2} % <-- your image file name
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\caption{$F_g$ vs $m_2$ graphed}
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\label{fig:m2graph}
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\end{figure}
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Note that again, there is a linear trend, implying
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\[
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F_g \propto m_2.
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\]
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Hence, the relationship can be described with the below equation, where $k_2$ is simply another constant:
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\begin{equation}
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F_g = k_2 \times m_2.
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\label{eq:m2}
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\end{equation}
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||
Looking at equation \ref{eq:m1} and \ref{eq:m2}, they can be safely combined into one equation, where there is yet another constant of proportionality $k_3$:
|
||
|
||
\begin{equation}
|
||
F_g = k_3 (m_1 \times m_2).
|
||
\label{eq:m1and2}
|
||
\end{equation}
|
||
|
||
Note, that when $m_1$ and $m_2$ are the same, equation \ref{eq:m1and2} simplifies to
|
||
|
||
\[
|
||
F_g = k_3 \times m^2.
|
||
\]
|
||
|
||
Looking at Table \ref{tab:grav_equal}, this derived relationship can be verified by graphing as follows:
|
||
|
||
|
||
|
||
\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 $m_1$ = $m_2$}
|
||
\label{fig:m1m2graph}
|
||
\end{figure}
|
||
|
||
|
||
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.\\\\
|
||
|
||
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 proportional 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 proportionality 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}
|
||
|
||
the force acting on the block can be calculated using Newton's Second Law, which can be used to calculate the frictional coefficients.
|
||
|
||
\section*{Conclusion}
|
||
|
||
\subsection*{Summary}
|
||
Revisiting the four objectives, the intended goals of this lab were to find the relationship between object masses, distance, and the resulting gravitational force,
|
||
thereby deriving NLUG and solving for $G$. The relationship between masses and the gravitational force was found in equation \ref{eq:m1and2}:
|
||
|
||
\begin{equation}
|
||
F_g = k_3 (m_1 \times m_2). \tag{\ref{eq:m1and2}}
|
||
\end{equation}
|
||
|
||
The second objective of finding the relationship between $F_g$ and the distance between the objects is captured in equation \ref{eq:finalEq}:
|
||
|
||
\begin{equation}
|
||
F_g = k_4 \times \frac{1}{r^2}. \tag{\ref{eq:finalEq}}
|
||
\end{equation}.
|
||
|
||
The third objective was met by calculating $G = 6.67 \times 10^{-11}\,\text{N\,m}^2\text{/kg}^2$, and the fourth objective was met
|
||
by deriving NLUG in equation \ref{eq:solvedNLUG}:
|
||
|
||
\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 \tag{\ref{eq:solvedNLUG}}
|
||
\end{equation}
|
||
|
||
The final equation \ref{eq:solvedNLUG} showed the relationship learned in this lab, that gravitation force is proportional to
|
||
the multiplied masses of the two objects involved divided by the square of the distance between their centers of masses.
|
||
|
||
|
||
\subsection*{Error Analysis}
|
||
|
||
One major source of error within this lab is the assumption that the simulation is entirely functional in simulating gravitational attraction.
|
||
In an real-world scenario, there are other external forces (friction, air resistance, electromagnetic forces) that can impact the gravitational attraction,
|
||
and if the simulation doesn't not properly isolate the gravitational force, the final relationship and the calculated gravitational constants can be off.
|
||
Additionally, the small nature of $G$ means that minor errors in any input variables or rounding errors have a profound impact on the
|
||
calculated force and gravitational constant. Moreover, the simulation only offers discrete input increments for distance and mass, whereas
|
||
the scale (particularly for distance) is free to move at any point in the screen. Hence, the distance intervals may be slightly inaccurate
|
||
as they depend on teh accurate placing of the two objects based on teh scale. Offering continuos distance inputs would offer a slight improvement.
|
||
Due to the relatively high number of trials conducted, these errors are relatively negligible in finding proportionalities; however, they can significantly impact the calculation of $G$.
|
||
Hence, $G$ was calculated using the default values for distance, ensuring that there would be no inaccuracy of the data point. Finally, the simulation
|
||
uses uniform spheres with a seemingly uniform density. This leads to an unproved generalization that NLUG works for two objects that do not have uniform density (or molecular makeup).
|
||
|
||
|
||
\subsection*{Critiques and Future Applications}
|
||
|
||
The primary critique regarding this lab is that the simulation itself does not render objects as they are in real life,
|
||
but rather uses NLUG to calculate the output. In other words, NLUG is derived from a simulation where NLUG is already built-in, which
|
||
nearly invalidates that the simulation as a valid source for data. Using physical objects comes with the additional limitation of human error
|
||
and external forces; however, they would more accurately showcase the true relationship of NLUG without a biased pre-coded simulation.
|
||
Additionally, adding more trials would help make the experimental setups more stable, and fixing the errors outlined in the Error Analysis
|
||
would greatly improve the validity of the experiment itself.\\
|
||
|
||
One specific application of NLUG is in calculating satellite orbits. Understanding NLUG allows engineers and physicists to
|
||
properly determine the force needed to maintain a satellite in stable orbit to ensure proper position for communication.
|
||
A second application is in planetary motion and astrophysics, where the same gravitational principles are used to
|
||
predict the trajectories of planets, moons, and comets. Additionally, NLUG can be used to calculate masses of planetary objects,
|
||
and proper understanding is required for accurate calculations.\\
|
||
|
||
This lab can be carried out differently and still illustrate teh same physics concepts.
|
||
For example, physical masses and a torsion balance can be used (similar to Cavendish) to directly measure the gravitational
|
||
attraction between small masses in a lab setting. Another method using two pendulums so measure minute changes in motion caused by gravitational attraction.
|
||
This will allow for a practical visualization of $F_G \propto 1/r^2$. Astronomical observations and using high precision force sensors are also alternatives,
|
||
although they would not be easily viable for a classroom lab setting.
|
||
|
||
|
||
\end{document}
|