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Kepler1 - Kepler's Second Law and Conserva tion of Angular...

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Kepler's Second Law and Conserva tion of Angular Momentum Kepler's Second Law is an example of the principle of conservation of angular momentum for planetary systems. We can make a geometrical argument to show how this works. Figure %: Small triangle swept out by planetary radius. Consider two points $P$ and $Q$ on the orbit of a planet, separated by avery small distance. Suppose that it takes a small time $dt$ for the planet to move from $P$ to $Q$. Because the line segment $\vec{PQ}$ is small, we can make the approximation that it is a straight line. Then $\vec{PQ}$, being the infinitesimal distance $dx$ over which the planet moved in time $dt$, represents the average velocity of the planet over that small range. That is $\vec{PQ} = \vec{v} $. Now consider the area swept out in this time $dt$. It is given by the area of the triangle $SPQ$, which has height $PP'$ and base $r$. But it is also clear from that $PP' = |PQ|\sin\theta$. Thus the area swept out per time $dt$ is given by: \begin{equation} \frac{dA}{dt} = \frac{1} {2}\times r \times |PQ| \times \sin\theta = \frac{rv\sin\theta}{2} \end{equation} But Kepler's
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