Phys 325 Spring 2011 Lecture 10 - Physics 325 Lecture 10...

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Unformatted text preview: Physics 325 Lecture 10 Planetary Motion in a Central Force Field (cont.) Another consequence of the conservation of angular momentum is Keplers 2 nd law. Consider the area swept out by the radius vector in the time interval dt : The area swept out is 2 1 2 2 r d dA r rd (The picture is greatly exaggerated. The triangle on the right that is adjacent to the trajectory is infinitesimally small (base dr , height rd )). The rate at which the area is swept out is 2 2 1 1 2 2 2 dA d L r r dt dt (10.1) Since L is constant, the area is swept out at a constant rate (Equal areas in equal times). Returning to the equation for the radial velocity we derived in Lecture 9 (Equation 9.13), 2 ( ) eff dr r E U r dt we see that r equals zero, when eff E U . In the case of the inverse square central force, we solved for these turning points in Lecture 8. Generally speaking there are two such turning points, and we call them min r and max r (perigee and apogee, respectively, in Lecture 7). d r(t) r(t+dt) =r+dr rd d r(t) r(t+dt) =r+dr rd We can calculate the angular change in one complete cycle from min r through max r and back to min r : max min 2 2 r...
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Phys 325 Spring 2011 Lecture 10 - Physics 325 Lecture 10...

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