From 1d105bcac3037070b651c33d083634ed2d9e7b11 Mon Sep 17 00:00:00 2001 From: KeshavAnandCode Date: Sun, 16 Nov 2025 12:43:11 -0600 Subject: [PATCH] Added masses relationship --- .../{ => images}/ExperimentalSetup.pdf | Bin .../images/Force vs Mass m1 m2.pdf | Bin 0 -> 39895 bytes .../{ => images}/Force vs m1.pdf | Bin .../images/Force vs m2.pdf | Bin 0 -> 28782 bytes PhysicsGravitationLabReport/main.pdf | Bin 273410 -> 341388 bytes PhysicsGravitationLabReport/main.tex | 55 +++++++++++++++++- 6 files changed, 52 insertions(+), 3 deletions(-) rename PhysicsGravitationLabReport/{ => images}/ExperimentalSetup.pdf (100%) create mode 100644 PhysicsGravitationLabReport/images/Force vs Mass m1 m2.pdf rename PhysicsGravitationLabReport/{ => images}/Force vs m1.pdf (100%) create mode 100644 PhysicsGravitationLabReport/images/Force vs m2.pdf diff --git a/PhysicsGravitationLabReport/ExperimentalSetup.pdf b/PhysicsGravitationLabReport/images/ExperimentalSetup.pdf similarity index 100% rename from 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--git a/PhysicsGravitationLabReport/main.tex b/PhysicsGravitationLabReport/main.tex index d6d42ff..c5278d6 100644 --- a/PhysicsGravitationLabReport/main.tex +++ b/PhysicsGravitationLabReport/main.tex @@ -16,6 +16,7 @@ \usepackage{placeins} \usepackage{float} +\graphicspath{{images/}} % ---------- Word-style margins ---------- @@ -399,18 +400,66 @@ $m_1$ and $F_g$ can be graphed as follows. 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: \[ -F_g \propto m_1 +F_g \propto m_1. \] -Hence, the relationship can be described witht eh below equation, where $k_1$ is simply a constant: +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:grav_force} +\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. + \section*{Error Analysis}