CENTRE

, or Center, in a general sense, signifies a point equally remote from the extremes of a line, plane, or solid; or a middle point dividing them so that some certain effects are equal on all fides of it.

Centre of Astraction, or Gravitation, is the point to which bodies tend by gravity; or that point to which a revolving planet or comet is impelled or attracted, by the force or impetus of gravity.

Centre of a Bastion, is a point in the middle of the gorge, where the capital line commences, and which is usually at the angle of the inner polygon of the figure. Or it is the point where the two adjacent curtains produced intersect each other.

Centre of a Circle, is the point in the middle of a circle, or circular figure, from which all lines drawn to the circumfereuce are equal.

Centre of a Conic Section, is the middle point of any diameter, or the point in which all the diameters intersect and bisect one another.

In the ellipse the centre is within the figure; but in the hyperbola it is without, or between the conjugate hyperbolas; and in the parabola it is at an infinite distance from the vertex.

Centre of Conversion, in Mechanics, a term first used by M. Parent, and may be thus conceived: Suppose a stick laid on stagnant water, and then drawn by a thread sastened to it, so that the thread always makes the same angle with the stick, either a right angle or any other; then it will be found that the stick will turn about one point of it, which will be immoveable; and this point is termed the centre of conversion.

This effect arises from the resistance of the fluid to the stick partly immersed in it. And if, instead of the body thus floating on a fluid, the same be conceived to be laid on the surface of another body; then the resistance of this plane to the stick will always have the same effect, and will determine the same centre of conversion. And this resistance is precisely what is called friction, so prejudicial to the effects of machines.

M. Parent has determined this centre in some certain cases, with much laborious calculation. When the thread is sastened to the extremity of the stick, he found that the distance of the centre from this extremity would be nearly 13/20 of the whole length. But when it is a surface or a solid, there will be some change in the place of this centre, according to the nature of the figure. See Mem. of the Acad. of Sciences, vol. 1, pa. 191.

Centre of a Curve, of the higher kind, is the point where two diameters meet.—When all the diameters meet in the same point, it is called, by Sir Isaac Newton, the general centre.

Centre of a Dial, is the point where its gnomon or stile, which is placed parallel to the axis of the earth, meets the plane of the dial; and from hence all the hour-lines are drawn, in such dials as have centres, viz, all except that whose plane is parallel to the axis of the world; all the hour-lines of which are parallel to the stile, and to one another, the centre being as it were at an infinite distance.

Centre of an Ellipse, is the middle of any diameter, or the point where all the diameters intersect.

Centre of the Equant, in the Old Astronomy, is a point in the line of the aphelion; being as far distant from the centre os the eccentric, towards the aphelion, as the sun is from the same centre of the eccentric towards the perihelion.

Centre of Equilibrium, is the same with respect to bodies immersed in a fluid, as the centre of gravity is to bodies in free space; being a certain point, upon | which if the body or bodies be suspended, they will rest in any position. To determine this centre, see Emmerson's Mechanics, prop. 92, pa. 134.

Centre of Friction, is that point in the base of a body on which it revolves, into which if the whole surface of the base, and the mass of the body were collected, and made to revolve about the centre of the base of the given body, the angular velocity destroyed by its friction would be equal to the angular velocity destroyed in the given body by its friction in the same time.—See Vince on the Motion of Bodies assected by friction, in the Philos. Trans. 1785.

Centre of Gravity, is that point about which all the parts of a body do in any situation exactly balance each other. Hence, by means of this property, if the body be supported or suspended by this point, the body will rest in any position into which it is put; as also that if a plane pass through the same point, the segments on each side will equiponderate, neither of them being able to move the other.

The whole gravity, or the whole matter, of a body may be conceived united in the centre of gravity; and in demonstrations it is usual to conceive all the matter as really collected in that point.

Through the centre of gravity passes a right line, called the diameter of gravity; and therefore the intersection of two such diameters determines the centre. Also the plane upon which the centre of gravity is placed, is called the plane of gravity; so that the common intersection of two such planes determines the diameter of gravity.

In homogeneal bodies, which may be divided lengthways into similar and equal parts, the centre of gravity is the same with the centre of magnitude. Hence therefore the centre of gravity of a line is in the middle point of it, or that point which bisects the line. Also the centre of gravity of a parallelogram, or cylinder, or any prisin whatever, is in the middle point of the axis. And the centre of gravity of a circle or any regular sigure, is the same as the centre of magnitude.

Also, if a line can be so drawn as to divide a plane into equal and similar parts, that line will be a diameter of gravity, or will pass through the centre of gravity; and it is the same as the axis of the plane. Thus the line drawn from the vertex and perpendicular to the base of the isosceles triangle, is a diameter of gravity; and thus also the axis of an ellipse, or a parabola, &c, is a diameter of gravity. The centre of gravity of a segment or arc of a circle, is in the radius or line perpendicularly bisecting its chord or base.

Likewise, if a plane divide a solid in the same manner, making the parts on both sides of it perfectly equal and similar in all respects, it will be a plane of gravity, or will pass through the centre of gravity. Thus, as the intersection of two such planes determines the diameter of gravity, the centre of gravity of a right cone, or spherical segment, or conoid, &c, will be in the axis of the same.

Common Centre of Gravity of two or more bodies, or the different parts of the same body, is such a point as that, if it be suspended or supported, the system of bodies will equiponderate, and rest in any position. Thus, the point of suspension in a common balance beam, or sseelyard, is the centre of gravity of the same.

Laws and Determination of the Centre of Gravity.

1. In two equal bodies, or masses, the centre of gravity is equally distant from their two respective centres. For these are as two equal weights suspended at equal distances from the point of suspension; in which case they will equiponderate, and rest in any position.

2. If the centres of gravity of two bodies A and B be connected by the right line AB, the distances AC and BC from the common centre of gravity C, are reciprocally as the weights or bodies A and B; that is, AC : BC :: B : A.

See this demonstrated under the article Balance.

Hence, if the weights of the bodies A and B be equal, their common centre of gravity C will be in the middle of the right line AB, as in the foregoing article. Also since A : B :: BC : AC, therefore ; whence it appears that the powers of equiponderating bodies are to be estimated by the product of the mass multiplied by the distance from the centre of gravity; which product is usually called the momentum of the weights.

Further, from the foregoing proportion, by composition it will be A + B : A :: AB : BC, or A + B : B :: AB : AC. So that the common centre of gravity C of two bodies will be found, if the product of one weight by the whole distance between the two, be divided by the sum of the two weights. Suppose, for example, that A = 12 pounds, B = 4lb, and AB = 36 inches; then 16 : 12 :: 36 : 27 = BC, and consequently AC = 9, the two distances from the common centre of gravity.

3. The Common Centre of Gravity of three or more given bodies or points A, B, C, D, &c, will be thus determined.—If the given bodies lie all in the same straight line AD; by the last article, find P the centre of gravity of the two A and B, and Q the centre of gravity of C and D; then, considering P as the place of a body equal to the sum of A and B, and Q as the place of another body equal to both C and D, sind S the common centre of gravity of these two sums, viz A + B collected in P, and C + D united in Q; so shall S be the common centre of gravity of all the four bodies A, B, C, D. And the same for any other number of bodies, always considering the sum of any number of them as united or placed in their common centre of gravity, when found.

Otherwise, thus. Take the distances of the given bodies from some fixed point as V, calling the distance VA = a, VB = b, VC = c, VD = d, and the distance of the centre of gravity VS = x; then SA = x-a, SB = x-b, SC = c-x, SD = d-x, and by the nature of the lever A.―(x-a) + B.―(x-b) = C.―(c-x) + D.―(d-x); hence Ax+Bx+Cx+Dx=Aa+ Bb+Cc+Dd, and the distance sought; which therefore is equal to the | sum of all the momenta, divided by the sum of all the we ghts or bodies.

Or thus. When the bodies are not in the same straight line, connect them with the lines AB, CD; then, as before, find P the common centre of A and B, and Q the common centre of C and D; then, conceiving A and B united in P, and C and D united in Q, find S the common centre of P and Q, which will again be the centre of gravity of the whole.

Or the bodies may be all reduced to any line VAB &c, drawn in any direction whatever, by perpendiculars BB, CC, &c, and then the common centre S in this ine, found as before, will be at the same distance from V as the true centre S is; and consequently the perpendicular from S will pass through S the real centre.

4. From the foregoing general expression, viz, , for the centre of gravity of any system of bodies, may be derived a general method for finding that centre; for A, B, C, &c, may be considered as the elementary parts of any body, whose sum or mass is M = A + B + C &c, and Aa, Bb, Cc, &c, are the several momenta of all these parts, viz, the product of each part multiplied by its distance from the fixed point V. Hence then, in any body, find a general expression for the sum of the momenta, and divide it by the content of the body, so shall the quotient be the distance of the centre of gravity from the vertex, or from any other fixed point, from which the momenta are estimated.

5. Thus, in a right line AB, all the particles which compose it may be considered as so many very small weights, each equal to x., which is therefore the fluxion of the weights, or of the line denoted by x. So that the small weight x. multiplied by its distance from A, viz x, is xx. the momentum of that weight x.; that is, xx. is the fluxion of all the momenta in the line AB or x; and therefore its fluent (1/2)x2 is the sum of all those momenta; which being divided by x the sum of all the weights, gives (1/2)x or (1/2)AB for the distance of the centre of gravity C from the point A; that is, the centre is in the middle of the line.

6. Also in the parallelogram, whose axis or length AB = x, and its breadth DE = b; drawing de parallel and indefinitely near DE, the areola dDEe = bx. will be the fluxion of all the weights, which multipl ed by its distance x from the point A, gives bxx. for the fluxion of all the momenta, and consequently the fluent (1/2)bx2 is the sum of all those momenta themselves; which being divided by bx the sum of all the weights, gives (1/2)x = (1/2)AB for the distance of the centre C from the extremity at A, and is therefore in the middle of the axis, as is known from other principles.

And the process and conclusion will be exactly the same for a cylinder, or any prism whatever, making b to denote the area of the end or of a transverse section of the body.

7. In a Triangle ABC; the line AD drawn from one angle to bisect the opposite side, will be a diameter of gravity, or will pass through the centre of gravity; for if that line be supported, or conceived to be laid upon the edge of something, the two halves of the triangle on both sides of that line will just balance one another, since all the parallels EF &c to the base will be bisected, as well as the base itself, and so the two halves of each line will just balance each other. Therefore, putting the base BC = b, and the axis or bisecting line AD = a, the variable part AS = x; then, by similar triangles AD : BC :: AS : EF, that is a : b :: x : bx/a = EF; which, as a weight, multiplied by x., gives bxx./a for the fluxion of the weights; and this again multiplied by x = AS, the distance from A, gives bx2x./a for the fluxion of the momenta; the fluent of which, or bx3/3a divided by bx2/a the fluent for the weights, gives (2/3)x = (2/3)AS for the distance of the centre of gravity from the vertex A in the triangle AEF; and when x = AD, then (2/3)AD is the distance of the centre of gravity of the triangle ABC.

The Same Otherwise, without Fluxions.—Since a line drawn from any angle to the middle of the opposite side passes through the centre of gravity, therefore the intersection of any two of such lines, will be that centre: thus then the centre of gravity is in the line AD; and it is also in the line CG bisecting AB; it is therefore in their intersection S. Now to determine the distance of S from any angle, as A, produce CG to meet BH parallel to AS in H; then the two triangles AGS, BGH are mutually equal and similar; for the opposite angles at G are equal, as are the alternate angles at H and S, and at A and B, also the side AG = BG; therefore the other sides BH, AS are equal. But the triangles CDS, CBH are similar, and the side CB = 2CD, therefore BH or its equal AS = 2DS, that is AS = (2/3)AD, the same as was found before. And in like manner CS = (2/3)CG. |

8. In a Trapezium. Divide the figure into two triangles by the diagonal AC, and find the centres of gravity E and F of these triangles; join EF, and find the common centre G of these two by this proportion, ABC : ADC :: FG : EG, or ABCD : ADC :: EF : EG.

In like manner, for any other sigure, whatever be the number of sides, divide it into several triangles, and find the centre of gravity of each; then connect two centres together, and find their common centre as above; then connect this and the centre of a third, and find the common centre of these; and so on, always connecting the last found common centre to another centre, till the whole are included in this process; so shall the last common centre be that which is required.

9. In the Parabola BAC. Put AD = x, BD = y, and the parameter = p. Then, by the nature of the figure, px = y2, and 2y = 2√px; hence 2x.√px is the fluxion of the weights, and 2xx.√px is the fluxion of the momenta; then the fluent of the latter divided by that of the former, or (4/5)x5/2p divided by (4/3)x3/2p, gives (3/5)x = (3/5)AD, for AG, the distance of the centre of gravity G from the vertex A of the parabola.

10. In the Circular Arc ABD, considered as a physical line having gravity. It is manisest that the centre of gravity G of the arc, will be somewhere in the axis, or middle radius BC, C being the centre of the circle, which is considered as the point of suspension. Suppose F indefinitely near to A, and FH parallel to BC. Put the radius BC or AC = r, the semiarc AB = z, and the semichord AE = x; then is AH = x., and AF = z. the fluxion of the weights, and therefore CE X z. is the fluxion of the momenta. But, by similar triangles, AC or r : CE :: AF or z. : AH or x., therefore rx. = CE X z., and so rx. is also the fluxion of the momenta; the fluent of which is rx, and this divided by z the weight, gives the distance of the centre of gravity from the centre C of the circle; being a 4th proportional to the given arc, its chord, and the radius of the circle.

Hence, when the arc becomes the semicircle ABK, the above expression becomes IC2/IB or r2/(1.5708r)=r/(1.5708) = .6366r, viz a third proportional to a quadrant and the radius.

11. In the Circular Sector ABDC. Here also the centre of gravity will be in the axis or middle radius BC. Now with any smaller radius describe the concentric arc LMN, and put the radius AC or BC = r, the arc ABD = a, its chord AED = c, and the variable radius CL or CM = y; then as r : y :: a : (ay)/r = the arc LMN, and r : y :: c : cy/r = the chord LON; also, by the last article, the distance of the centre of gravity of the arc LMN is ; hence the arc LMN or ay/r multiplied by y. gives (ayy.)/r the fluxion of the weights, and this mul tipliedby cy/a the distance of the common centre of gravity, gives (cy2y.)/r the fluxion of the momenta; the fluent of which, viz (cy3)/(3r), divided by ay2/2r, the fluent of the weights, gives (2cy)/(3a) for the distance of the centre of gravity of the sector CLEN from the centre C; and when y = r, it becomes 2cr/3a = CG for that of the sector CABD proposed; being 2/3 of a 4th proportional to the arc of the sector, its chord, and the radius of the circle.

Hence, when the sector becomes a semicircle, the last expression becomes 4r2/3a = 2IC2/3IB or 2/3 of a 3d proportional to a quadrantal arc and the radius. Or it is equal to 4r/3p = .4244r from the centre C; where p=3.1416.

12. In the Cone ADB. Putting a = DC, b = area of the base AEB, and x = Dc any variable altitude; then as a2 : x2 :: b : bx2/<*>2 = area aeb; hence the flux- | ion of the weights is bx2x./a2, whose fluent, or the solid, is bx3/3a2; and the fluxion of the momenta is bx3x./a2, whose fluent is bx4/4a2; then this fluent divided by the former fluent gives (3/4)x or (3/4)Dc for the distance of the centre of gravity of the cone Dab, or (3/4)DC for that of the cone DAB below the vertex D.

And the same is the distance in any other pyramid. So that all pyramids of the same altitude, have the same centre of gravity.

13. In like manner are we to proceed for the centre of gravity in other bodies. Thus, the altitude of the segment of a sphere, or spheroid, or conoid, being x, a being the whole of that axis itself; then the distance of the centre of gravity in each of these bodies, from the vertex, will be as follows, viz, (4a-3x)/(6a-4x)x in the sphere or spheroid, (5/8)x in the semisphere or semispheroid, (2/3)x in the parabolic conoid, (4a + 3x)/(6a + 3x)x in the hyperbolic conoid.

14. To determine the Centre of Gravity in any Body Mechanically. Lay the body on the edge of any thing, as a triangular prism, or such like, moving it backward and forward till the parts on both sides are in equilibrio; then is that line just in, or under the centre of gravity. Balance it again in another position, to find another line passing through the centre of gravity; then the intersection of these two lines will give the place of that centre itself.

The same may be done by laying the body on an horizontal table, as near the edge as possible without its falling, and that in two positions, as lengthwise and breadthwise: then the common interfection of the two lines contiguous to the edge, will be its centre of gravity. Or it may be done by placing the body on the point of a style, &c, till it rest in equilibrio. It was by this method that Borelli found that the centre of gravity in a human body, is between the nates and pubis; so that the whole gravity of the body is collected into the place of the genitals; an instance of the wisdom of the creator, in placing the membrum virile in the part, which is the most convenient for copulation.

The same otherwise thus. Hang the body up by any point; then a plumb-line hung over the same point, will pass through the centre of gravity; because that centre will always descend to the lowest point when the body comes to rest, which it cannot do except when it falls in the plumb line. Therefore, marking that line upon it, and suspending the body by another point, with the plummet, to find another such line, the intersection of the two will give the centre of gravity.

Or thus. Hang the body by two strings from the same tack, but fixed to different points of the body; then a plummet, hung by the same tack, will fall on the centre of gravity.

In the 4th volume of the New Acts of the Academy of Petersburgh, is the demonstration of a very general theorem concerning centres of gravity, by M. Lhuilier; a particular example only of the general proposition, will be as follows: Let A, B, C, be the centres of gravity of three bodies; a, b, c their respective masses, and Q their common centre of gravity. Let right lines QA, QB, QC, be drawn from the common centre to that of each body, and the latter be connected by right lines AB, AC, and BC; then

.

Uses of the Centre of Gravity. This point is of the greatest use in mechanics, and many important concerns in life, because the place of that centre is to be considered as the place of the body itself in computing mechanical effects; as in the oblique pressures of bodies, banks of earth, arches of bridges, and such like.

The same centre is even useful in finding the superficial and solid contents of bodies; for it is a general rule, that the superficies or solid generated by the rotation of a line or plane about any axis, is always equal to the product of the said line or plane drawn into the circumference or path described by the centre of gravity. For example, it was found above at art 11, that in a semicircle, the distance of the centre of gravity from the centre of the circle, is 4r/3p; and therefore the path of that centre, or circumference described by it whilst the femicircle revolves about its diameter, is (2/3)r; also the area of the semicircle is (1/2)pr2; hence the product of the two is (4/3)pr3, which, it is well known, is equal to the solidity of the sphere generated by the revolution of the semicircle.

And hence also is obtained another method of finding mathematically the centre of gravity of a line or plane, from the contents of the superficies or solid gene<*> rated by it. For if the generated superficies or solid be divided by the generating line or plane, the quotient will be the circumference described by the centre of gravity; and consequently this divided by 2p gives the radius, or distance of that centre from the axis of rotation. So, in the semicircle, whose area is (1/2)pr2, and the content of the sphere generated by it (4/3)pr3; here the latter divided by the former is (8/3)r, and this divided by 2p gives 4r/3p for the distance of the centre of gravity from the axis, or from the centre of the semicircle. The property last mentioned, relative to the relation between the centre of gravity and the figure generated by the revolution of any line or plane, is mentioned by Pappus, in the preface to his 7th book; and father Guldin has more fully demonstrated it in his 2d and 3d books on the Centre of Gravity.

The principal writers on the centre of gravity are Archimedes, Pappus, Guldini, Wallis, Casatus, Carré, Hays, Wolfius, &c.

Centre of Gyration, is that point in which if the whole mass be collected, the same angular velocity will be generated in the same time, by a given force acting at any place, as in the body or system itself. This point differs from the centre of oscillation, in as much as in this latter case the motion of the body is produced by the gravity of its own particles, but in the case of the | centre of gyration the body is put in motion by some other force acting at one place only.

To determine the Centre of Gyration, in any body, or system of bodies composed of the parts A, B, C, &c, moving about the point S, when urged by a force f acting at any point P. Let R be that centre: then, by mechanics, the angular velocity generated in the system by the force f, is as , and, by the same, the angular velocity of the matter placed all in the point R, is ; then since these two are to be equal, their equation will give , for the distance of the centre of gyration sought, below the axis of motion.

Now because the quantity , where G is the centre of gravity, O the centre of oscillation, and b the whole body or sum of A, B, C, &c; therefore it follows that ; that is, the distance of the centre of gyration, is a mean proportional between those of gravity and oscillation.

And hence also, if p denote any particle of a body, placed at the distance d from the axis of motion; then is ; from whence the point R may be determined in bodies by means of Fluxions.

Centre of an Hyperbola, is the middle of the axis, or of any other diameter, being the point without the figure in which all the diameters intersect one another; and it is common to all the four conjugate hyperbolas.

Centre of Magnitude, is the point which is equally distant from all the similar external parts of a body. This is the same as the centre of gravity in homogeneal bodies that can be cut into like and equal parts according to their length, as in a cylinder or any other prism.

Centre of Motion, is the point about which any body, or system of bodies, moves, in a revolving motion.

Centre os Oscillation, is that point in the axis or line of suspension of a vibrating body, or system of bodies, in which if the whole matter or weight be collected, the vibrations will still be performed in the same time, and with the same angular velocity, as before. Hence, in a compound pendulum, its distance from the point of suspension is equal to the length of a simple pendulum whose oscillations are isochronal with those of the compound one.

Mr. Huygens, in his Horologium Oscillatorium, first shewed how to find the centre of oscillation. At the beginning of his discourse on this subject, he says, that Mersennus first proposed the problem to him while he was yet very young, requiring him to resolve it in the cases of sectors of circles suspended by their angles, and by the middle of their bases, both when they oscillate sideways and flatways; as also for triangles and the segments of circles, either suspended from their vertex or the middle of their bases. But, says he, not having immediately discovered any thing that would open a passage into this business, I was repulsed at first setting out, and stopped from a further prosecution of the thing; till being farther incited to it by adjusting the motion of the pendulums of my clock, I surmounted all difficulties, going far beyond Descartes, Fabry, and others, who had done the thing in a few of the most easy cases only, without any sufficient demonstration; and solving not only the problems proposed by Mersennus, but many others that were much more difficult, and shewing a general way of determining this centre, in lines, superficies, and solids.

In the Leipsic Acts for 1691 and 1714, this doctrine is handled by the two Bernoullis: and the same is also done by Herman, in his treatise De Motu Corporum Solidorum et Fluidorum.

It may also be seen in treatises on the Inverse Method of Fluxions, where it is introduced as one of the examples of that method. See Hayes, Carré, Wolfius, &c.

To determine the Centre of Oscillation, in any Compound Mass or Body MN, or of any System of Bodies A, B, C, &c.

Let MN be the plane of vibration, to which plane conceive all the matter to be reduced by letting fall perpendiculars to this plane from every particle in the body; a supposition which will not alter the vibration of the body, because the particles are still at the same distance from the axis of motion. Let O be the centre of oscillation, and G the centre of gravity; through the axis S draw SGO, and the horizontal line ST; then from every particle A, B, C, &c, let fall perpendiculars Aa and Ap, Bb and Bq, Cc and Cr, &c, to these two lines; and join SA, SB, SC; also draw Gm and On perpendicular to ST.

Now the forces of the weights A, B, C, to turn the body about the axis, are A . Sp, B . Sq,-C . Sr; and, by mechanics, the forces opposing that motion are A . SA2, B . SB2, C . SC2; therefore the angular motion generated in the system is . In like manner, the angular velocity which any body or particle p, situated in O, generates in the system, by its weight, is because of | the similar triangles SGm, SOn. But, by the conditions of the problem, the vibrations are performed alike in both these cases; therefore these two expressions must be equal to each other, that is , and consequently . But, by mechanics again, the sum of the forces A . Sp + B . Sq-C . Sr is equal the force of the same matter collected all into its centre of gravity G; and therefore , which is the distance of the centre of oscillation O below the axis of suspension.

Farther, because it was found under the article Centre of Gravity, that , therefore is the same distance of the centre of oscillation; where any of the products A . Sa, B . Sb, &c are to be taken negatively when the points a, b, &c lie above the point S, or where the axis passes through. Again, because, by Eucl. II 12 and 13, it is ; and because by Mechanics, the sum of the last terms is nothing, ; therefore the sum of the others, or ; where b denotes the body, or sum A+B+C &c of all the parts: this value then being substituted in the numerator of the 2d value of SO above-found, it becomes

From which it appears that the centre of oscillation is always below the centre of gravity, and that the difference or distance between them is .

It farther follows from hence, that ; that is, the rectangle SG.GO is always the same constant quantity, wherever the point of suspension S is placed, since the point G and the bodies A, B, &c, are constant. Or GO is always reciprocally as SG, that is GO is less as SG is greater; and the points G and O coincide when SG is infinite; but when S coincides with G, then GO is infinite, or O is at an infinite distance.

To find the Centre of Oscillation by means of Fluxions. From the premises is derived this general method for the centre of oscillation, viz, let x be the abscissa of an oscillating body, and y its corresponding ordinate or section; then will the distance SO of the centre of oscil- lation below the axis of suspension S, be equal to the fluent of yx2x divided by the fluent of yxx.. So that, if from the nature or equation of any given figure, the value of y be expressed in terms of x, or otherwise, and substituted in these two fluxions; then the fluents being duly found, and the one divided by the other, the quotient will be the distance to the centre of oscillation in terms of the absciss x.

But when the body is suspended by a very fine thread of a given length a, then the fluent of ―(a + x))2 . yx. divided by the fluent of ―(a + x) . yx. gives the distance of the same centre of oscillation below the point of suspension.

Ex. For example, in a right line, or rectangle or cylinder or any other prism, whose constant section is y, or the constant quantity a; then yx2x. is ax2x., whose fluent is (1/3)ax3; also yxx. is axx., whose fluent is (1/2)ax2; and the quotient of the former (1/3)ax3 divided by the latter (1/2)ax2, is (2/3)x for the distance of the centre of oscillation below the vertex in any such figure, namely having every where the same breadth or section, that is, at twothirds of its length.

In like manner the centre of oscillation is found for various figures, vibrating flatways, and are as they are expressed below, viz,

Nature of the Figure.When suspended by Vertex.
Isosceles triangle3/4 of its altitude
Common Parabola5/7 of its altitude
Any Parabola(2m+1)/(3m+1) X its altitude.

As to figures moved laterally or sideways, or edgeways, that is about an axis perpendicular to the plane of the figure, the finding the centre of oscillation is somewhat difficult; because all the parts of the weight in the same horizontal plane, on account of their unequal distances from the point of suspension, do not move with the same velocity; as is shewn by Huygens, in his Horol. Oscil. He found, in this case, the distance of the centre of oscillation below the axis, viz,

In a circle,3/4 of the diameter:
In a rectangle, susp. by one angle,2/3 of the diagonal:
In a parabola susp. by its vertex,5/7 axis + 1/3 param.
The same susp. by mid. of base,4/7 axis + 1/2 param.
In a sector of a circle(3 arc X radius)/(4 chord):
In a cone4/5 axis +(radius base2)/(5 axis):
In a sphereg + 2r2/5g, where r is
the radius, and the rad. added to the length of the thread.
See also Simpson's Fluxions, art. 183 &c.

To find the Centre of Oscillation Mechanically or Experimentally. Make the body oscillate about its point of suspension; and hang up also a simple pendulum of such a length that it may vibrate or just keep time with the other body: then the length of the simple pendulum is equal to the distance of the centre of oscillation of the body below the point of suspension.

Or it will be still better found thus: Suspend the body very freely by the given point, and make it vibrate | in small arcs, counting the vibrations it makes in any portion of time, as a minute, by a good stop watch; and let that number of oscillations made in a minute be called n: then shall the distance of the centre of oscillation be SO=(140850)/nn inches. For, the length of the pendulum vibrating seconds, or 60 times in a minute, being 39 1/8 inches, and the lengths of pendulums being reciprocally as the square of the number of vibrations made in the same time, therefore n2 : 602 :: 39 1/8 : 140850/nn the length of the pendulum which vibrates n times in a minute, or the distance of the centre of oscillation below the axis of motion.

Centre of Percussion, in a moving body, is that point where the percussion or stroke is the greatest, in which the whole percutient force of the body is supposed to be collected; or about which the impetus of the parts is balanced on every side, so that it may be stopt by an immoveable obstacle at this point, and rest on it, without acting on the centre of suspension.

1. When the percutient body revolves about a sixed point, the centre of percussion is the same with the centre of oscillation; and is determined in the same manner, viz, by considering the impetus of the parts as so many weights applied to an inflexible right line void of gravity; namely, by dividing the sum of the products of the forces of the parts multiplied by their distances from the point of suspension, by the sum of the forces. And therefore what has been above shewn of the centre of oscillation, will hold also of the centre of percussion when the body revolves about a fixed point. For instance, that the centre of percussion in a cylinder is at 2/3 of its length from the point of suspension, or that a stick of a cylindrical figure, supposing the centre of motion at the hand, will strike the greatest blow at a point about two-thirds of its length from the hand.

2. But when the body moves with a parallel motion, or all its parts with the same celerity, then the centre of percussion is the same as the centre of gravity. For the momenta are the products of the weights and celerities; and to multiply equiponderating bodies by the same velocity, is the same thing as to take equimultiples; but the equimultiples of equiponderating bodies do also equiponderate; therefore equivalent momenta are disposed about the centre of gravity, and consequently in this case the two centres coincide, and what is shewn of the one will hold in the other.

Centre of Percussion in a fluid, is the same as out of it.

Centre of a Parallelogram, the point in which its diagonals intersect.

Centre of Pressure, of a fluid against a plane, is that point against which a force being applied equal and contrary to the whole pressure, it will just sustain it, so as that the body pressed on will not incline to either side.—This is the same as the centre of percussion, supposing the axis of motion to be at the intersection of this plane with the surface of the fluid; and the centre of pressure upon a plane parallel to the horizon, or upon any plane where the pressure is uniform, is the same as the centre of gravity of that plane. Emerson's Mechanics, prop. 91.

Centre of a Regular Polygon, or Regular Body, is the same as that of the inscribed, or circumscribed circle or sphere.

Centre of a Sphere, is the same as that of its generating semicircle, or the middle point of the sphere, from whence all right lines drawn to the superficies, are equal.

Centring of an Optic Glass, the grinding it so as that the thickest part be exactly in the middle.

Cassini the younger has a discourse expressly on the necessity of well centring the object glass of a large telescope, that is, of grinding it so as that the centre may fall exactly in the axis of the telescope. Mem. Acad. 1710.

Indeed one of the greatest difficulties in grinding large optic glasses is, that in figures so little convex, the least difference will throw the centre two or three inches out of the middle. And yet Dr. Hook remarks, that though it were better the thickest part of a long object glass were exactly in the middle, yet it may be a very good one when it is an inch or two out of it. Philos. Trans. N° 4.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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CELESTIAL Globe
CELLARIUS (Christopher)
CENTAURUS
CENTESM
CENTRAL
* CENTRE
CENTRIFUGAL Force
CENTRIPETAL Force
CENTROBARICO
CENTRUM
CEPHEUS