C E N
( i79 )
C E N
derating towards the right; to determine the 'Point V, from whence the Sum of all the Weights being fufpended, the Pre- ponderation Jball continue the fame as in their former
Find the Momentum wherewith the Weights c and d preponderate towards the right ; fince the Momentum of the Sum of the Weights to be fufpended in F, is to be equal to it, the Momentum now found, will be the Fac- tum of C F into the Sum of the Weights : This, there- fore, being divided by the Sum of the Weights, the Quo- tient will be the Diilance C F, at which the Sum of the Weights is to be fufpended, that the Preponderation may continue the fame as before.
6. To find the Centre of Gravity in a Parallelogram and parallelepiped. Draw the Diagonals AD and EG, (Fig. 1<S.) likewise CB and HE ; fince each Diagonal, AB andCB divides the Parallelogram into two equal Parts, each paffes thro the Centre of Gravity ; confecjuently, the Point of Intcrfeclion I, is the Centre of Gravity of the Pa- rallelogram. In like manner, fince both the Plane C B F H and" ADGE, divide the Parallelepiped into two equal Parts, each paffes thro its Centre of Gravity ; fo that the common Interfeclion I K, is the Diameter of Gravity, the Middle whereof is the Centre.
After the fame manner may the Centre of Gravity be found in Prifms and Cylinders 5 it being the middle Point of the right Line that joins the Centres of Gravity of their oppofite Bafes.
7. In regular Polygons, the Centre of Gravity is the fame with the Centre of the circumfcrib'd Parallelogram.
8. To find the Centre of Gravity of a Cone and a Pyra- mid. The Centre of Gravity of a Cone, is in its Axis AC, (Fig. 17.) If then AP = ^,Pj> = ^.r, the Weight in the fame Cone is prx* dx : 2 a' ; and therefore its Mo- mentum firx'dx. 1 a 1 . Hence the Sum of the Momenta prx*: %a x 5 which divided by the Sum of the Weights prx* : 6 a 1 , gives the Diilance of the Centre of Gravity of the Portion AMN, from the Vertex A = <Sfl* frx* : Sa z f> r.v*=| x — ? 4 Kp : wherefore, the Centre of Gra- vity of the entire Cone, is diflant from the Vertex, \ of AC. And in the fame manner is found rhc Diilance of the Cen- tre of Gravity from the Vertex of the Pyramid {. A C.
9. To determine the Centre of Gravity in a 'Triangle BAC, (Fig. 18.) Draw the right Line AD, biffecling the Bafe EC in D ; fince aBAD = AD AC, each may be di- vided into the fame Number of Weights, apply'd in the fame manner on each fide to the common Axis A D : So that the Centre of Gravity of the ABA C, will be in A D. To determine the precife Point in that, let A D = K,BC = J, P = .r, MN=jv ; then will AP:MN=AD :BC X : y = a : h. Hence, y = h x : a. Draw A E = c perpendicular to B C 5 then A D : A E = A P : A Qj and therefore, AQj= ex: a, and Q_? === c d x : a. Whence, the Momentum y x d x = c bx z dx : a', and fyx dx = c bx* : 3 a/ ; which Sum di- vided by the Area of the Triangle AMN = cij! 1 :2i( t , gives the Diilance of the Centre of Gravity from rhe Vertex = iabx s : 3abx* — }x. If then for x, be fubllituted a, the Diilance of the Centre of Gravity of the A, from the Vertex, will be found f a.
10. For the Centre of Gravity in a Parabola, (Fig. 19.) Its Diilance from the Vertex A, is found the Space A F. In a cubical Paraboloid, the Diilance of the Centre from the Vertex, is=^AP. In a Biquadratic Paraboloid, £AP. In a Surdefolid Paraboloid, ~ T A P. In the Exterior Para- bola AST, the Centre of Gravity is at the Diilance AL. In the Cubical Paraboloid, t AQ. In a Biquadratic Pa- raboloid, i A Q: In a Surdcfolidal Paraboloid, f A Q^
ir. the Centre of Gravity in the Arch of a Circle, is dif- tant from the Centre of the Arch by a Line, which is a third Proportional to the Quadrant and the Radius. In a Sector of a Circle, the Diilance of the Centre of Gravity from fhe Centre of the Circle, is to the Diilance of the Centre of Gravity of the Arch, as 2 to 3.
For the Centre of Gravity of Segments, Lines, Para- bolic Conoids, Spheroids, truncated Cones, &c. as being Cafes more operofe, and at the fame time more out of the way ; we refer to Wo/fius's Elem. Mathef. Tom. I.
11. To determine the Cenrre of Gravity in any Body me- chanically. Lay the given Body H I, (Fig. 20.) on an extended Rope, or the Edge of a triangular Prifm F G, bringing it this and that way, till the Parts on either Side arc in JEquilibrio ; rhe Plane whofe Side is KL, paffes thro the Centre of Gravity. Balance it again on the feme, only changing its Situation : then will the Cord MN, p a f s t i, ro the Centre of Gravity ; fo that the Inter- jection of the two Lines M N and K L, determines the
il)\ °. m the Surface of the Body rcquir'd.
, J, lme ma y ne c ' one by l a >mg the Body on a hori- Sv -, • ' ( as neiU ' the Ed £ e as is poffible, without its tailing) in two Pofitions, lengthwife and breadthwife : the common ImerfeSion of the two Lines contiguous to the
Edge, will be its Centre of Gravity. Or, it may be done by laying the Body on the Point of a Style c5?c till it reft 1,1 JEqmlibrio. 'Twas by this Method, Borelli found the Centre of Gravity in an human Body, to be be- tween the Nates and Pubis ; fo that the whole Gravity of the Body is there collected, where Nature has plac'd the Genitals : An Inllance of the Wifdom of the Creator, in placing the Membrum Virile in that Place, which of all others is the moll convenient for the Affair of Copulation.
13.' Every Figure, whether fuperficial or foiid, generated
by the Motion of a Line, or Figure, is equal to the Fac-
| turn of the generating Magnitude, multiply'd into the
Way of its Centre of Gravity, or the Line its Centre of
' Gravity defcribes.' See the Demonllrarion hereof, under
the Article Centro -Baric Method.
The preceding elegant Theorem, is look'd on as one of the noblcil Geometrical Difcoveries made in the lair Age ; and is the Foundation of the Methcdus Centro-ba- rica. Pappus, indeed, gave the firll Hint long ago ; but it was the Jefuit Guldinus that brought it to Maturity. Leibnitz, fhews it will hold, if the Axis, or Centre, be connnually chang'd during rhe generating Motion : The Corollaries arc too numerous to be here detail'd.
Centre of Motion, is a Point round which one or more heavy Bodies, that have one common Centre of Gravity, revolve, v.g. If the Weights P and Q_ (Tab. 3teehamcks, Fig. ii.) revolve about the Point M, fo as when P de- fcends, Q_ afcends, N is laid to be the Centre of Mo- tion. See Motion.
'Tis demonstrated in Mcchanicks, that the Diilance I N, of the Centre of Gravity of any particular Weight, from the common Centre of Gravity, or the Centre of Motion N, is perpendicular to the Line of Direction I p.
Centre of Ofcitlation, a Point wherein, if the whole Gtavity of a compound Pendulum be collected, the feveral Ofcillations will be perform'd in the fame Time as before : Sec Oscillation.
Hence, its Diilance from the Point of Sufpenfion, is equal to the Length of a fimple Pendulum, whofe Ofcillations are Ifochronal with thofe of the compound one. See Pen- "
DULUM.
Laws of the Centre of Ofcillation.
1. If feveral Weights DFHB, (Tab. Mcchanicks, Fig. 21.) whofe Gravity isfuppos'd colleBed in the Points DEHB, conftautly retain the fame T)i fiance between tbemfelves and the Point of Sufpenfion A ; and the Pendulum thus compounded, perform its Ofcillations about A ; the D; fiance of the Centre of Ofcillation O, from the Point of Sufpen- fion O A, will be had by multiplying the feveral Weights into the Squares of the Difiances, and dividing the Aggre- gate by the Sum of the Momenta of their Weight.
2. To determine the Centre of Ofcillation in a right Line AB, (Fig. 23.) LetAE = ff,AD=i, then will the infi- nitely fmall Particle D P = dx, the Momentum of its Weight xdx ; confequently the Diilance of the Centre of Ofcil- lation in the Part AD, from the Point of Sufpenfion A = fx z d x :fxdx— I X s : £x z —y X. If then for x be fubllituted a, the Diilance of the Centre of Ofcillation in the right Line AB — y&. In this manner is found the Centre of Ofcillation of a Wire, ofcillating about one of its Extremes.
3. To determine the Centre of Ofcillation of the Rectangle R1HS, (Fig. 24.) fufpended in the middle Point A, of the Side RI, and ofcillating about its Axis RI. Let RI=SH=a,AP, then will ¥p =dx, and the Element of the Area ; confequently one Weight ~adx, andits Mo- mentum axdx. Wherefore, fax l dx : fax dx^l-ax*
- iax'=lx, indefinitely expreffes the Diftance of the
Centre of Ofcillation, from the Axis of Ofcillation in the Segment RCDI. If then for x be fubllituted the Alti- tude of the whole Rectangle RS = £, we (hall have the Diilance of the Centre of Ofcillation from the Axis = f I.
For the Centre of Ofcillation in an Equi crural Triangle, ofcillating about its Axis, parallel to its Bafe, its Diilance from the Vertex, is found = J of the Altitude of the Triangle. Of an Equicrural Triangle ofcillating about its Bafe, its Diilance from the Vertex is found =i the Altitude of the Triangle.
For the Centre of Ofcillation in an Equicrural Tri- angle, fufpended by an inflexible Thread, void of Gra- vity, and ofcillating about its Axis parallel to its Bafe ; its Diilance from the Vertex, is found = i the Altitude of the Triangle.
For the Centres of Ofcillation of 'Parabolas, and Curves of the like kind, ofcillating about their Axis, parallel to their Bafes, they are found as follows.
In the Apollonian Parabola, rhe Diftance of the Centre from the Axis, =j of its Diameter.
In a Cubical Paraboloid, the Diilance of the Centre of Ofcillation from the Axis, = fs- of the Diameter.
In a Biquadratic Paraboloid, the Diilance of the Centre from the Axis, == ? T of the Diameter.
In
