The elements of astronomy; a textbook . ed,first, that the linear velocity (usu-ally denoted by V) is always in- FlG 131 versely proportional to Sb, the Linear and Angular Velocities. perpendicular drawn from S upon AB, produced if necessary : secondly, that the angular velocity(ordinarily denoted by <o) at any point of the orbit is inverselyproportional to the square of AS, the radius vector. In every case of motion under central force we may say, therefore : I. The areal velocity (acres per second) is constant. II. The linear velocity (m i le s per second) varies inversely as the dis-tanc
The elements of astronomy; a textbook . ed,first, that the linear velocity (usu-ally denoted by V) is always in- FlG 131 versely proportional to Sb, the Linear and Angular Velocities. perpendicular drawn from S upon AB, produced if necessary : secondly, that the angular velocity(ordinarily denoted by <o) at any point of the orbit is inverselyproportional to the square of AS, the radius vector. In every case of motion under central force we may say, therefore : I. The areal velocity (acres per second) is constant. II. The linear velocity (m i le s per second) varies inversely as the dis-tance from the centre of force to the bodys line of motion at the moment. III. The angular velocity (degrees per second) varies inversely asthe square of the radius vector. These three statements are not independent laws, but only geomet-rical equivalents for each other. They hold good regardless of thenature of the force, requiring only that when it acts it act directlytowards or from the centre, so as to be directed always along the line. 374 APPENDIX. [§ 502 of the radius vector. It makes no difference whether the force varieswith the square or the logarithm of the distance; whether it is increas-ing or decreasing, attractive or repulsive, continuous or intermittent,provided only it be always central. 503. Proof of the Law of Inverse Squares, from KeplersHarmonic Law (supplementary to Art. 253). —For circular orbitsthe proof is very simple. From equation (&), Art. 250, wehave for the first of two planets, in which/j is the central force (measured as an acceleration),and i\ and £x are respectively the planets distance from the sunand its periodic a second planet, hDividing the first equation by the second, we get /i - Tl x (U /2 r2 ViBut by Keplers third law t2 r3ti : t} = r}: r23; whence, f-2 = -2s; t 2substituting this value of -^ in the preceding equation, we have f\ _ n v r2s _ r,f2~ X 3 „. 2 > , fi :f2 = r2 : r22, — which is the law of inverse squares
Size: 1765px × 1416px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjec, booksubjectastronomy
