Analytical mechanics for students of physics and engineering . (II) we get r2u=/*, (III) where h is a constant. The following properties, which aredirect consequences of equation (III), are common to allmotions in central fields of force. (1) The radius vector sweeps over equal areas in equalintervals of time. When the radius vector turns through an angle dd it sweepsover an area equal to \r>rdd; therefore the rate at whichthe area is described equals 1 J 1. 1, -r- — = -r-co = - h = constant. 2 dt 2 2 (2) The angular velocity of the particle varies inverselyas the Bquare of the distance of
Analytical mechanics for students of physics and engineering . (II) we get r2u=/*, (III) where h is a constant. The following properties, which aredirect consequences of equation (III), are common to allmotions in central fields of force. (1) The radius vector sweeps over equal areas in equalintervals of time. When the radius vector turns through an angle dd it sweepsover an area equal to \r>rdd; therefore the rate at whichthe area is described equals 1 J 1. 1, -r- — = -r-co = - h = constant. 2 dt 2 2 (2) The angular velocity of the particle varies inverselyas the Bquare of the distance of the particle from the centerof force. This is evident from equation (III). (3) The linear velocity of the particle varies inversely asthe Length of the perpendicular which is dropped upon thedirection of the velocity from the center of force. It was shown on page 87 that VCOSd) where v is the linear velocity and the angle which the velocity makes with a line perpendicular to the radius vector. i denote the length of the perpendicular dropped from. MOTION OF A PARTICLE 285 the center of force upon the direction of the velocity; fcheDit is evident from Fig. 128 that cos= —r Substituting this value of cos in the preceding equation weobtain pv (x) = — J r- Fig. 128. or v= — = -• (1\) V P (4) The angular momentum of the particle with respectto the center remains constant. This result is obtained at once by multiplying both sidesof equation (III) by m, the mass of the particle. Thus mr2w = mh, but mr2co = Iu. Therefore loo = mh = constant. 219. Equation of the Orbit. — The general equation of theorbit is found by eliminating t between equations (I) and(III). The analytical reasoning which follows does not needfurther explanation: ^L _ dr d$ _ dr dt~ dd dt~ U dd . = h drr2 dd - 3 dd hdu = - h —fdd [by (Hi)] 286 ANALYTICAL MECHANICS where u = -. Therefore d-rdt2 , d2u dd~ do1 dt _ ,„ 0d2u — h-U—r-- Substituting this value of -^and the value of — , which maybe obta
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