C E N
The Effect of the Centrifugal Force is fuch, that a Bo- dy, oblig'd to defcribe a Circle, defcribes the largeft it poffibly can : a greater Circle being as it were lefs cir- cular, and lefs dii;ant from a right Line than a (mall one. A Body therefore fuffers more violence, and exerts its Centrifugal Force more when it defcribes a little Circle, than when a large one.
'Tis the fame in other Curves as in Circles $ for a Curve, whatever it be, may be efteem'd as compos'd of an infi- nity of Arches of infinitely final) Circles, all defcrib'd on different Radii; fo that 'tis thofe Places where the Curve has ihe greateft Curvity, that the little Arches are molt circular : Thus, in the fame Curve, the Centrifugal Force of the Body that defcribes it, varies according to the feve- ral Points wherein it is found.
CENTRIPETAL Force, is that Power whereby a moveable Body, impell'd in the right Line AG, (Fig. 24.) is dtawn out or its reftilinear Motion, to proceed in a Curve : The Centripetal Force, therefore, is as the right Line DE to AB$ fuppofmg the Arch AE infinitely final], Hence, the Centripetal and Centrifugal Forces are equal.
Lams of Central Forces,
1. The following Rule, for which we are oblig'd to the Marquis dc I' Hipital, opens at once all the Myfteries of Central Forces: Suppofe a Body of any determinate Weight to move uniformly round a Centre with any certain Velo- city j find from what height it mull have fallen to acquire that Velocity : then, as the Radius of the Circle it de- fcribes, is to double that Height, fo is its Weight to its Centrifugal Ferce. Hence, 'tis eafy to infer, that,
z. if two Bodies, equal in we ght, defcribe Peripheries of unequal Circles in equal Times, their Central Forces are as their Diameters AB, and HL. And hence, if theCentral Forces of two Bodies, defcribing Peripherics of two une- qual Circles, be as their Diameters, they pafc over the Jame in equal Times
3. The Central Free of a Body moving in the Periphe- ry of a Circle, is as tie Square of the infinitely Jmall Arch A E, divided by the Diameter AB. Since then a Body, by an equable Motion in equal Times, defcribes equal Arches A E; the Central Force wherewith the Bo- dy is impell'd in the Periphery of the Circle, is constantly the fame.
4. If two Bodies defcribe different Peripheries by an cquab'e Morion, their Central Forces are in a Ratio, compounded 0. the duplicate Ratio of their Celerities, and the reciprocal one of tflfiir Diamcrers. Hence, if the Ce- lerities be equal, the Central Forces will be reciprocally as their Diameters 3 and if the Diameters AB and H L be equal, r. e. if each Moveable ] roceed in the fame Peri- phery, but with unequal Celerities, the Central Forces will be in a duplicate Ratio of the Velockies.
If the Central Forces ot the two Bodies moving in diffe- rent Peripherics be equal, the Diameters of .he Circle A B, and HL, will be in a duplicate Ratio of the Celerities.
5. It two Bodies, moving in unequal Peripheries, be aclcd on by the fime Central Force, the Time in the larger is to that in the fmailer, in a tubduplicate Ratio of the greater Diameter A B, to the lef HL 5 wherefore, T 1 : t* = D; d : That is, the Diameters of the Circles in whofe P-ripheries thofr Bodie- are acled on by the fame Central Force, are in a duplicate Ratio of the Times. HeiK:; aifo the Times wherein like Peripheries or Arches are run over by Bodies impell'd by the fame Central Force, are in proportion to their Velocities.
The Central Forces are in a Ratio, compounded of the direct Ratio ot the Diameters, and the reciprocal one of the Squares of the Times, by the entire Peripheries.
6. If the Times wherein the Bodies are carry'd thro the fame entire Peri hcries, or fimilar Arches, be as the Diamcrers of the Circles, the Central Forces are reci.ro- cally as the 'ame Diameters.
7. If a Body move uniformly in the Periphery of a Cir- cle, with the Velocity it acquires by falling the Height A L i the Central Force will be to the Gravuy, as double the Altitude A L, to the Radius C A. If therefore the Gravit of the Body be call'd G, the Centrifugal Force will be z fl L. G : C A.
8. If a heavy Body move equably in the Periphery of a Circle, and with the Velocity which it acquires by falling the height A L, equal to half the Radius; the Central Force will be equal to the Gravity. And again, if the Om- tral Force be equal to the Gravity, 'tis carry'd in the Peri- phery of the Circle, with the fame Gravity which it acquires in falling a height equal to half the Radius.
9. If the Central Force be equal to the Gravity, the Time it takes up in the entire Periphery, is to the Time of the Defcent thro half the Radius, as the Periphery to the Radius.
10. If two Bodies move in unequal Peripheries, and with an unequal Velocity, which is reciprocally in a fubduplicate
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Ratio of the Diameters; the Central Forces are in a 0*. plicate Ratio of the Diftances from the Centre of the For. ces, taken reciprocally.
II. If two Bodies move in unequal Peripherics, with Celerities that are reciprocally as the Diameters : their Central Forces will be reciprocally as the Cubes of their Distances from the Centre of the Forces.
I*. If the Velocities of two Bodies moving in unequal Peripherics, be reciprocally in a fubduplicate Ratio of the Diameters; the Times wherein they pals the whole Pe- riphery, or fimilar Arches, are reciprocally in a triplicate Ratio of the Diflances from the Centre of the Forces : Wherefore, ,f the Central Forces be reciprocally in a du- pl.c ite Ratio of the Dilknces from the Centre, the Times wherein the entire Peripheries, or fimilar Arches, are pafs'd over, are reciprocally in a triplicate Ratio of the Diftances.
13. It a Body move in a Curve Line, in fuch manner as that the Radius C B, Fig. drawn from it to the fix'd Point C, plac d m the fame Plane, defcribes Areas BAG, ECE, iSc. proportional to the Time Time, it is follicited towar Force.
14. If a Body proceed according to the Direction of the
or equal in any given the Point C by a Centrijwtel
are proportional to the Times.
15. However the Central Forces differ from one ano- ther, they may be compar'd together; for they are always in a Ratio compounded of the Ratio of the Quantities of Matter in the revolving Bodies, and the Ratio of the Dif- tances from the Centre; and alfo in an inverfe Ratio of the Squares of the periodical Times. If then you multiply the Quantity of Matter in each Body by its Diitance from the Centre, and divide the Produfl by the Square of the perio- dical Time, the Quotients of the Divifion will be to one ano'herin the faid compound Ratio, that is, as the Central Forces.
16. When the Quantities of Matter are equal, the Dif- tances themfelves mull be divided by the Squares of the periodical Times, to determine the Proportion of the Cen- tral Votxs : In that Cafe, if the Squares of the periodi- cal Times arc to one another, as the Cubes of the Diftan- ces, the Quotients of the Divifions, as well as the Central Forces, will be in an inverfe Ratio of the Squares of the Diilances.
17. When the Force by which a Body is carry'd towards a Point is not every where the fame, but is either increas'd or diminifh'd, in proportion to the Diitance from the Cen- tre; feveral Curves will thence arife in a certain propor- tion. If the Force decreafes, in an inverfe Ratio of the Squares of the Diftances from that Point, the Body will defcribe an Elliffls; which is an oval Curve, in which there ire two Points call'd the Feci, and the Point towards which the Force is direfled falls into one of them : So that in every Revolution, the Body once approaches to and once recedes from it. The Circle alfo belongs to that fort of Curve., and fo in that Cafe the Body may alfo de- lcribe a Circle. The Body may alfo (by fuppofing a great- er Celerity m it) defcribe the two remaining Conic Sec- tions, ib. the 'Parabola and HyperUla-Cvrves, which do not return into themfelves : On the contrary, if the Force increafes with the Diftance, and that in a Ratio of the Dii- tance it felf, the Body will again defcribe an Elligis- but the Point to which the Force is direded is rhe Centre of the Elhpfe; and the Body, in each Revolution, will twice approach to, and again twice recede from that Point. In this Cafe alfo, a Body may move in a Circle, for the Rea- fon abovemention'd. See Orbit, Planet, and Projectile
Central Rale, is a Rule, or Method difcover'd by our Countryman, Mr. Thomas 'Baker, Rcflor of Nvmptoa in Devon, whereby to find the Centre of a Circle - defign'd to cut the Parabola in as many Points, as an Equation to be conftructed hath real Roots.
Its principal Ufe is in the Conftruftion of Equations • and he has apply'd it with good Succefs as far as Biquadra- tics. See Construction, and Equation.
The Rules are thefe,
r+
| = } = CD.
La. PPt_, t± _L
4 T «UX 4 LLX'
Or by Contraction, avoid Fractions.
LL
= i = DE.
becaufe L = i, as is fuppofed, to
5-1-2
- ■ iJ> + }ptp+4 p?+;r=d=ET>.
CENTRO-S«r)>c Method of Guldinus, in Mechanicks, is a Method of meafuring, or determining the Quantity Bbb of
