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Physics LibreTexts

9.2: Kepler's Second Law from Conservation of Angular Momentum

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Figure 9.1.png
FIGURE IX.1

In figure IX.1, a particle of mass m is moving in some sort of trajectory in which the only force on it is directed towards or away from the point O. At some time, its polar coordinates are (r,θ). At a time δt later these coordinates have increased by δr and δθ.

Using the formula one half base times height for the area of a triangle, we see that the area swept out by the radius vector is approximately

δA=12r2δθ+12rδθδr.

On dividing both sides by δt and taking the limit as δt0, we see that the rate at which the radius vector sweeps out area is

˙A=12r2˙θ.

But the angular momentum is , mr2˙θ and since this is constant, the areal speed is also constant. The areal speed, in fact, is half the angular momentum per unit mass.


This page titled 9.2: Kepler's Second Law from Conservation of Angular Momentum is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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