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