\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