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Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>

src/Advanced/Geometry of Masses/parts/1 balance 2d.tex

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+\section{Balancing a plane}
+
+\definition{}
+Consider a massless two-dimensional plane. \par
+Affix a finite number of point masses to this plane. \par
+We will call the resulting object a \textit{two-dimensional system of masses:}
+
+
+\begin{center}
+	\begin{tikzpicture}[scale = 0.5]
+
+		%\draw[
+		%	line width = 0mm,
+		%	pattern = north west lines,
+		%	pattern color = blue,
+		%]
+		%	(1, 0)
+		%	-- (0.5, 0.866)
+		%	-- (-0.5, 0.866)
+		%	-- (-1, 0)
+		%	-- (-0.5, -0.866)
+		%	-- (0.5, -0.866)
+		%	-- cycle;
+		%\draw[
+		%	line width = 0.5mm,
+		%	blue
+		%]
+		%	(1, 0)
+		%	-- (0.5, 0.866)
+		%	-- (-0.5, 0.866)
+		%	-- (-1, 0)
+		%	-- (-0.5, -0.866)
+		%	-- (0.5, -0.866)
+		%	-- cycle;
+		%\fill[color = blue] (0, 0) circle[radius=0.3];
+
+
+		\fill[color = black]
+			(-3, 3) circle[radius = 0.5]
+			node[above] at (-3, 3.5) {$m_1$ at $(x_1, y_1)$};
+
+		\fill[color = black]
+			(-5, -1.5) circle[radius = 0.4]
+			node[above] at (-5, -1.0) {$m_2$ at $(x_2, y_2)$};
+
+		\fill[color = black]
+			(3, -3) circle[radius = 0.35]
+			node[above] at (3, -2.5) {$m_3$ at $(x_3, y_3)$};
+
+		\draw[line width = 0.5mm]
+			(-7.5, -4.2)
+			-- (6, -4.2)
+			-- (6, 5)
+			-- (-7.5, 5)
+			-- cycle;
+
+	\end{tikzpicture}
+\end{center}
+\vspace{5mm}
+
+
+
+\problem{}
+Show that any two-dimensional system of masses has a unique center of mass. \par
+\hint{
+	If a plane balances on a pin, it does not tilt in the $x$ or $y$ direction. \par
+	See the diagram below.
+}
+\begin{center}
+	\begin{tikzpicture}[scale = 0.5]
+
+		% Horizontal
+		\draw[line width = 0.5mm, dotted, gray] (-3, 3) -- (-3, -5);
+		\draw[line width = 0.5mm, dotted, gray] (-5, -1.5) -- (-5, -5);
+		\draw[line width = 0.5mm, dotted, gray] (3, -3) -- (3, -5);
+		\draw[line width = 0.5mm, dotted, gray] (0, 0) -- (0, -5);
+		\draw[line width = 0.5mm] (-7, -5) -- (6.5, -5);
+
+		\fill[color = gray] (-3, -5) circle[radius = 0.3];
+		\fill[color = gray] (-5, -5) circle[radius = 0.3];
+		\fill[color = gray] (3, -5) circle[radius = 0.3];
+
+		\draw[line width = 0.25mm, pattern=north west lines]
+			(0, -5) -- (-0.6, -6) -- (0.6, -6) -- cycle;
+
+
+		% Vertical
+
+		\draw[line width = 0.5mm, dotted, gray] (-3, 3) -- (8, 3);
+		\draw[line width = 0.5mm, dotted, gray] (-5, -1.5) -- (8, -1.5);
+		\draw[line width = 0.5mm, dotted, gray] (3, -3) -- (8, -3);
+		\draw[line width = 0.5mm, dotted, gray] (0, 0) -- (8, 0);
+		\draw[line width = 0.5mm] (8, 4) -- (8, -4);
+
+		\fill[color = gray] (8, 3) circle[radius = 0.3];
+		\fill[color = gray] (8, -1.5) circle[radius = 0.3];
+		\fill[color = gray] (8, -3) circle[radius = 0.3];
+
+		\draw[line width = 0.25mm, pattern=north west lines]
+		(8, 0) -- (9, -0.6) -- (9, 0.6) -- cycle;
+
+
+		\draw[
+			line width = 0mm,
+			pattern = north west lines,
+			pattern color = blue,
+		]
+			(1, 0)
+			-- (0.5, 0.866)
+			-- (-0.5, 0.866)
+			-- (-1, 0)
+			-- (-0.5, -0.866)
+			-- (0.5, -0.866)
+			-- cycle;
+		\draw[
+			line width = 0.5mm,
+			blue
+		]
+			(1, 0)
+			-- (0.5, 0.866)
+			-- (-0.5, 0.866)
+			-- (-1, 0)
+			-- (-0.5, -0.866)
+			-- (0.5, -0.866)
+			-- cycle;
+		\fill[color = blue] (0, 0) circle[radius=0.3]
+			node[above] at (0, 1) {Pivot};
+
+		\fill[color = black]
+			(-3, 3) circle[radius = 0.5]
+			node[above] at (-3, 3.5) {$m_1$ at $(x_1, y_1)$};
+
+		\fill[color = black]
+			(-5, -1.5) circle[radius = 0.4]
+			node[above] at (-5.5, -1.0) {$m_2$ at $(x_2, y_2)$};
+
+		\fill[color = black]
+			(3, -3) circle[radius = 0.35]
+			node[above] at (3, -2.8) {$m_3$ at $(x_3, y_3)$};
+	\end{tikzpicture}
+\end{center}
+
+\begin{solution}
+	\begin{equation*}
+		x_0 = \frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^n m_i}
+		\qquad
+		y_0 = \frac{\sum_{i=1}^n m_i y_i}{\sum_{i=1}^n m_i}
+	\end{equation*}
+\end{solution}
+
+\vfill
+\pagebreak
+
+\vfill
+\pagebreak