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% Header: title centered, partner left, date right
\begin{center}
    \textbf{\large Experimentally Deriving Newton’s Universal Law of Gravitation by Measuring Mass, Distance, and G in a Simulation}
\end{center}

\vspace{1em}

\noindent
\textbf{Name:} Keshav Anand
\hfill
\textbf{Date:} 16 November 2025
% ---------- MAIN DOCUMENT ----------


\section*{Objectives}

\begin {itemize}
    \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.
    \item Use the same simulation to determine the dependence of the gravitational force on the distance between the two masses.
    \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.)
    \item Determine an Equation for the Universal Law of Gravitation based on your data, using only symbols.
\end {itemize}

\section*{Introduction}

\subsection*{Background}
The concept of gravitational force has been misunderstood for most of history,
commonly attributed to divine or supernatural causes until Sir Isaac Newton published his groundbreaking \textit{Principia}.
In the Principia, Newton asserted that every mass exerts an attractive force on every other mass,
 a phenomenon described by Newton's Universal Law of Gravitation (NLUG). 
This law states that the magnitude of the gravitational force between two masses is

\begin{equation}
F_g = 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 proportionality that has been calculated to be 
\[
G = 6.674 \times 10^{-11}\ \text{N}\,\text{m}^2\,\text{kg}^{-2}.
\]

In the scientific community, NLUG is treated as an absolute truth, and many important discoveries and applications
rely on its accuracy. From engineering to astrophysics, NLUG has profound importance, and its validity
is vital to the functioning of scientific advancement. 

\subsection*{Purpose}

This lab aims to use a computer simulation 
to verify NLUG by deriving the relationship between gravitational force, masses of objects, and the distance
between them. First, a gravitation simulation will be used to derive the relationship between objects'
masses and the gravitational force. Then, the same process will be repeated with comparing gravitational
force to the distance between objects. Data collected from these two setups will be used to (hopefully) re-establish
the relationship of proportionality proposed by Newton, and the collected data will also solve for the 
universal Gravitational constant, $G$.


\subsection*{Research Problem}
The research objective for this project is to verify the gravitational relationship between two objects and verify the
constant $G$. The primary problem of investigation is that theoretical mathematics often fails to adequately capture
a true relationship in the real world. Moreover, using physical objects and tools of measurements can result in
unwanted noisy data and is limited by the precision of measurement. Ergo, a simulation bridges this gap, 
allowing for an accurate verification of NLUG.




\section*{Methodology}
\subsection*{Materials and Resources}

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.

Within the simulation, the simulated materials include
\begin{itemize}
    \item Adjustable Mass, $m_1$
    \item Adjustable Mass, $m_2$
    \item 10 Meter Scale
    \item Automatic Force Scale to Measure Gravitational Attraction
    \item Two simulated people holding $m_1$ and $m_2$ from colliding into each other due to gravitation
\end{itemize}


\subsection*{Experimental Setup}

\begin{figure}[h!] % h! = “here” placement
    \centering
    \includegraphics[width=0.7\textwidth]{ExperimentalSetup} % <-- your image file name
    \caption{Experimental setup for the gravity simulation}
    \label{fig:friction_setup}
\end{figure}

Note that in Figure \ref{fig:friction_setup}, all inputs (independent variables) are denoted in blue, whereas outputs (dependent variables) are denoted in red


\subsection*{Procedure}
\begin{enumerate}
    \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
    \item Set the mass of objects 1 and 2 to exactly 100 kg
    \item Set the force values to scientific notation, and uncheck the option for masses of constant size
    \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 $m_1$
      \subitem Mass values for $m_1$ (kg): 50, 100, 250, 500, 750, 1000
    \item Reset the simulation as detailed by steps 1-3
    \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 $m_1$
      \subitem Mass values for $m_2$ (kg): 50, 100, 250, 500, 750, 1000
    \item Reset the simulation as detailed by steps 1-3
    \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 $m_1$
      \subitem Mass values for $m_1$ and $m_2$ (kg): 50, 100, 250, 500, 750, 1000
    \item Reset the simulation as detailed by steps 1-3
    \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 $m_1$
    \subitem Position values for $m_1$ (m): 0, 2, 4, 6, 8
    \item Reset the simulation as detailed by steps 1-3
    \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 $m_1$
    \subitem Position values for $m_1$ (m): 10, 8, 6, 4, 2

\end{enumerate}

Note that the above steps require the following raw data the be collected at each datapoint

\begin{itemize}
  \item Position of the center of mass of $m_1$, (m)
  \item Position of the center of mass of $m_2$, (m)
  \item Mass of $m_1$, (kg)
  \item Mass of $m_2$, (kg)
  \item Force on $m_1$ by $m_2$, (N)
  \item Force on $m_2$ by $m_1$, (N)
\end{itemize}

\section*{Results}



\subsection*{Raw Data}

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{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{$m_1$ (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 & 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}

\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}

\begin{table}[H]
\centering
\caption{Force between two masses while varying $x_1$.}
\label{tab:grav_dist1}
\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 & 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}

\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}

\subsection*{Discussion and 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.
\\\\
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 with the below equation, where $k_1$ is simply a constant:

\begin{equation}
F_g = k_1 \times m_1
\label{eq:m1}
\end{equation}

Applying the same logic to Table \ref{tab:grav_m2} yields the following relationship between $F_g$ and $m_2$


\begin{figure}[h!] % h! = “here” placement
    \centering
    \includegraphics[width=0.7\textwidth]{Force vs m2} % <-- your image file name
    \caption{$F_g$ vs $m_2$ graphed}
    \label{fig:m2graph}
\end{figure}

Note that again, there is a linear trend, implying 

\[
F_g \propto m_2.
\]

Hence, the relationship can be described with the below equation, where $k_2$ is simply another constant:

\begin{equation}
F_g = k_2 \times m_2.
\label{eq:m2}
\end{equation}

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}
