CENTKIFUOAI FOKCE.] MECHANICAL PHILOSOPHY. DYNAMICS. 



a right-angled plane triangle, since the angle B is i 

 semicircle ; therefore (Euc. 8, VI). 



Fig. 146. 



M 



741 



is in a 



/. AE 



:AA' 



AB . . _, _ -.~ 



AA' = 2AS-- 2AJfi ~~AS~ P 



Now 2 A E represents the accelerating force at S, or, 

 taken in an opposite direction, it represents the centri- 

 fugal force /, and A B represents the velocity v in the 



curve ; consequently the centrifugal force /= , where r 



radius. 



If as usual be made to stand for the number 

 314159, <fcc., the whole circumference of the circle will 

 be 2 v r ; therefore, calling the whole time of describing 

 the circumference that U, the periodic time I, and 

 remembering that the uniform velocity v is equal to the 

 whole space divided by the whole time, we have 



2-irr Wr 



T'J- t - ' 



In the foregoing reasoning we are required to take ( 

 so many secmulu ; but the unit of time may be regarded 

 as much smaller than this as we please ; and thus the 

 error of confounding the arc A B with its chord may be 

 entirely removed, BO that we may conclude rigorously 

 that 



The centrifugal force in a circle varies as the radius 

 divided by the square of the time of describing it : thus, 



for another circle, we should have F 



where /, F stand for the centrifugal or centripetal forces 

 at the respective centres of circles of radii r, R, and in 

 which the periodic times are t, T. If R=r, that is, if 

 the circles are equal, the centrifugal forces F, / are in- 

 versely as the squares of the times T, t. 



An interesting application of these results U to the 

 determination of the centrifugal force at different places 

 on the surface of the earth, from knowing the time of one 

 rotation on its axis. But in studying this application, 

 the student will not fail to observe that the circular path 

 of a body, on the surface of the rotating earth, is not de- 

 scribed under the same circumstances as the circular path 

 is considered to be described in the present article. Here 

 there ia no solid matter interposed between the circulat- 

 ing body and the central force ; there is nothing to pre- 

 vent its falling to the centre at any point of its orbit ; 

 it is kept always at the same distance from the centre, 

 simply because the force pulling it towards that centre is 

 exactly balanced by that driving it from it : the centri- 

 fugal force is just sufficient to deprive the body of all 

 weight or pressure towards the centre. With the earth 

 it is different : if a perforation be bored in the earth, 

 beneath a body on its surface, the body will fall down it; 

 because the centrifugal force, driving it from the centre, 

 is less than the attracting, or centripetal, force pulling it 

 the other way. 



CENTRIFUGAL FORCE AT THE EARTH'S SUR- 

 FACE. The earth, by its diurnal rotation, carries round, 

 with a uniform velocity, every point oil its surface in 

 86164 seconds. At the equator, the radius R is about 

 20922000 feet ; therefore the centrifugal force F at the 

 equator is 



_, 4*rR 4jrX 20922000 



F = ~T* X - 86164* ~ = ' 11124 feei 



As this force opposes the force of gravity, it follows 

 that if the earth had no rotation on its axis that is, if 

 no centrifugal force existed gravity at the equator, in- 

 stead of being what it really is, namely, g = 32 '088 feet, 

 would be 0=3-)- !! 1245, and thus the weight of a body 



there would be a "anao P ar * more than it actually is. 



The ratio of G to F being 



32119 : 1112 or 289 : 1 nearly ; 



Now every parallel to the equator being carried round 

 in the same time T, as the equator itself, by represent- 

 ing the centrifugal force iu the parallel whose latitude ia 

 I, and radius r, by /, we shall have 



/ = r_ A/=F r = G cogj (2) 



Since, as is evident, r=R cos I. 



The force of gravity is not diminished by the whole 

 of the centrifugal force, except at the equator ; because 

 in any parallel PAP' (Fig. 147) this force acts, not in 

 the direction I" p wholly opposed to the force of gravity, 

 but in the direction P' p'. If, therefore, we decompose 

 the force P' p in the perpendicular directions P' p, 1" q, 

 the former component, being directly opposed to the 

 force of gravity, and the latter component being a force 

 acting tangentially, and therefore urging the particle P 

 towards the equator, we shall have 

 Fig. 147. 



Force opposing gravity, P' 

 cos. I. 



P' p' cos. p P' p'-, f 



Hence (2) the expression for the diminution of gravity, 

 in consequence of centrifugal force, at latitude I, is 



- 



Diminution of equatorial gravity G in lat. 



/-t 



cos. 2 l. 

 289 



Consequently the amount of diminution varies as the 

 square of the cosine of the latitude. 



The other component P' q of the centrifugal force at 

 P, being tangential, tends to drive the particles of the 

 revolving body from the region of the poles to that of the 

 equator, and to cause the body to assume the figure of 

 an oblate spheroid, which is the figure that the earth has 

 assumed: the expression for the tangential force is 



Tangential force, P'2=/sin. l=<jgjj s i n - & cos - ' 



G 



"678 sin " 27) 



which therefore varies as the sine of twice the latitude. 

 The force thus tending to accumulate matter about the 

 equatorial regions, is obviously greatest at lat. 45, for 

 there 21=1. 



