Prop. 1. To find the centre of gravity of two given bodies, geometrically. used to denote that point in which, | cession to two different points of if a particle of matter were placed, the body, and the point of interthe action of each body upon it section of these two directions will would be equal, and where it will show the centre required. therefore remain in equilibrio. Let M and m represent the masses of the two bodies, and d their distance from each other. Put x for the distance of the point of equal attraction from M. and y the distance of the same from m; then by the laws of attraction √m+ √ M CENTRE of a Circle, is that point in a circle which is equally distant from every point of the circumference; and, if more than two equal lines can be drawn from any point within a circle to the circumfer ence, that point will be the centre. CENTRE of Equilibrium, is the same with respect to bodies in mersed in a fluid as the centre of gravity is to bodies in free space. CENTRE of Friction, is that point in the base of a body on which it revolves, into which, if Divide the line joining them in the inverse ratio of the bodies. Thus, if a be two pounds, one pound, and they be three feet asunder, then will the centre of gravity be one foot from a and two from b. gravity of a triangle. Bisect any two of its sides, draw lines to the opposite angles, and the intersection of these lines will be the centre of gravity; for, as the triangle balances itself upon each of the lines, it must also balance itself upon the point where they meet. Prop. 3. To find the centre of gravity of any rectilinear figure. Divide it into triangles, find their centres of gravity, join them, and the result will be a new figure, having fewer sides. Repeat this operation till the figure be reduced to one triangle or to two. If to one, find its centre of gravity: If to two find both, and divide their distance inversely as the triangles: The ultimate point in either case is the centre required. General Laws of the Centre of 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 de Prop. 1. To find the centre of stroyed by its friction would be any number of bodies placed in a equal to the angular velocity de-right line. stroyed in the given body by its friction in the same time. CENTRE of Gravity of any body, or system of bodies, is that point upon which the body or system of bodies, acted upon only by the force of gravity, will balance itself in all positions; hence it follows, that if a line or plane pass ing through the centre of gravity be supported, the body or system will be also supported. Multiply each body by its distance from some fixed point in the line, and divide the sum of the products by the sum of the bodies, the result is the distance of the centre from the point. Prop. 2. If perpendiculars be drawn from any number of bodies to a given plane, the sum of the products of each body into its respective perpendicular distance from the plane, is equal to the product of the sum of all the bodies into the perpendicular distance of their common centre of gravity from the plane. To find the Centre of Gravity of bodies, mechanically, it is only necessary to dispose the body successively in two positions of equilibrium, by the aid of two forces in From this may be derived a gevertical directions, applied in suc-neral method for finding the centre centre of gravity, and p = 3.14159, &c., we have LG = flu. x s 1. Now in the case of an area, s of gravity of a system of bodies. For Au. 2 yxx __ fiu. y xx = 、 Au. 2yx flu. yx 2. In the case of a simple curve fiu. 2 xx flu. 2 x 3. For the solid, mated. But, in order to and the... LG Prop. 3. To find the centre of gravity of a body considered as an area, solid, surface of a solid, or curve line. Let ALV (Plate II. fig. 3.) be any curve, RL the axis, in which the centre of gravity must lie; for as it bisects every ordinate IF in N, the parts on each side RL will always balance each other, and therefore the body will balance itself upon RL, and consequently the centre of gravity must be somewhere in that line. Put LN, IN = y, IL = %, and draw PQ parallel to IF; then if we consider this body to be made up of an indefinite number of corpuscles, and multiply each into its distance from PQ, the sun of all the products divided by the sum of all the corpuscles, or by the whole body LG, will give us the distance of the centre of gravity from L; as is shown above in the preceding proposition. Now, to get the sum of all the products, we must first have the Auxion of the sum, the fluent of which will be the sum itself. Puts for the fluxion of the body at the distance LN, or a, then will * s be the fluxion of the sum of all the products; also, s the fluxion of the sum of all the corpuscles. Hence, taking G to represent the Pyx, LG fu. py2 x 4. For the simple surface of the solid, s=py ≈, ..LG Au.py xx = flu. yxi fu.py z Au. yz The centre of gravity of the following bodies, putting generally a middle of the base, is for the line joining the vertex and A plain triangle a from the A right cone a vertex. A circular sector, as arch: chord =3 radius, to the distance of centre of gravity from the centre. = The altitude of the segment of a sphere, or spheroid or conoid, being represented by x, and the whole axis by a, the distance of the centre of gravity in each of these bodies from the vertex will be The sphere or spheroid 4a3n 6u-4x collected, a given force applied at a given distance would produce the same angular velocity in the same time as if the bodies were disposed at their respective distances. This point differs from the centre of oscillation only in this, that in the latter case the motion is produced by the gravity of the body; but in the former, the body is put in motion by some other force acting at one place only. To determine the Centre of Gyration. P X SD2 &c. Let the whole niass be collected at rest. in R; and the accelerating force point in the axis of suspension of CENTRE of Oscillation is that upon D is PX SD2 (A+B+C+, &c.) × SR3 AX SA2+ B x SB+, &c. (A+B+C+, &c.) x SR2 Whence SR= AX SA2+BX SB2+, &c. A+B+C+, &c.; Consequently, if s be the fluxion of the body at the distance x from the axis, SRV Au. 22's S a vibrating body, in which, if all the matter of the system were collected, any force applied there would generate the same angular velocity in a given time, as the same force at the centre of gravi ty, the parts of the system revolv ing in their respective places. Let several bodies oscillate about a point of suspension, as if the mass of each were concentrated into points referred to the same plane perpendicular to the axis of motion. Then the gravity of each of them may be decomposed into two forces, of which the one passing through the centre of suspenand so the other, perpendicular in sion is destroyed by its resistance; direction to the former, is alone efficacious in moving the body or system. Now gravity tends to impress the same velocity upon the points in the vertical direction; This, in the case of a right line, which velocity we shall denote by becomes In a circle, or cylinder, revolving about the axis, radius X. The periphery of a circle, about the diameter, radius X . A wheel with a very thin rim, revolving about its axle, radius. g, and by m, n, p, the sines of the angles, which the supposed inflexible bars, joining the bodies with the centre of suspension, form with the perpendicular. Drawing lines parallel to this perpendicular, and each equal to g, they will represent the accelerating forces of the bodies, or the spaces which they would describe in the first unit of time, if they were left to them-1 selves. But because of the obliquity of these forces upon the inflexible bars, if rectangles be constructed, the spaces run over will be only the sides of those rectangles which are at right angles to the bars; and as the angles have for their sines m, n, p, we shall have these respectively equal to gm, gn, and gp. Hence it follows, that the bodies taken separately, move with different veloci ties. But if we suppose them connected together, so that they all perform their vibrations in the same time, the velocity of some will be augmented while that of others will be diminished; and as the aggregate of the forces which solicit the system is always the same, it is necessary that the sum of the motions lost should be equal to that of the motions gained. ga (A + B + C) hr Aa2+Bb2+C c2• Let us represent, by A, B, C, the To ascertain the actual position masses of the three bodies, by a, of the point whose invariable conb, c, their distances from the point nection with the system does not of suspension, and by a, B, y, the change its velocity, let a be its initial velocities which they lose; distance from the centre of susthe quantities of motion lost will pension, and S the sine of the anbe A a, BB, C7, which must be ingle which the inflexible rod that equilibrio; therefore, the sum of their momentums taken with regard to the point of suspension is nothing; and as these respective distances from that point are a, b, c, we shall have retains it to that point makes with from x= x (A+B+C) g h r x a =gs, s AaBb2 + Cc2 (A+B+C) h• That the point sought may be the centre of oscillation, it is not merely necessary that these two velocities be equal in the first instant, they must continue so in every instant of the descent; there fore a remaining the same, this equation should hold whatever be the position of the point sought, and that of the centre of gravity, x= A a2 + B b2 + C c2 (A+B+C) h When the percutient body re volves about a fixed point, the cen tre of percussion is the same with the centre of oscillation. For instance, 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. CENTRE of Pressure, is that point against which a force being applied equal and contrary to the whole pressure, will sustain it, so as that the body pressed on will not incline to either side. CENTRE of Position, in Mechanics, denotes a point of any body, The same kind of reasoning ap- or system of bodies, so selected, plies exactly, however many the that we may properly estimate number of particles may be there- the situation and motion of the bofore, to find the centre of oscilla-dy, or system, by those of this point. tion of a system of particles or of bodies, we must multiply the weight of each of them by the square of its distance from the point of suspension, and divide the sum of these products by the weights multiplied by the distance of the centre of gravity from the centre of suspension: this quotient expresses the distance of the centre of oscillation from the point of suspension measured on the continuation of the line joining the centre of gravity and that point. Call S the point of suspension, O the centre of oscillation, or SO the distance of the centre of oscillation from the point of suspension; also lets be the fluxion of the body at the distance r; then the above formula becomes SO= Au. x2 s flu. x 8 As an example, let it be proposed to find the centre of oscillation of a right line, or cylinder, suspended at one end. In this case Au. x2 xx8 SO = CENTRE of spontaneous Rotation, is that point which remains at rest the instant a body is struck, or about which the body begins to revolve. CENTRIFUGAL Force, is that by which a body revolving about a centre or about another body, has a tendency to recede from it. CENTRIPETAL Force, is that by which a body is perpetually urged towards a centre, and there. by made to revolve in a curve instead of a right line. CENTRIPETATION, a term used by Sir Richard Phillips to indicate the tendency which bodies or planets, and parts of planets, have to fall or move towards their centres, which tendency he ascribes to the orbicular and rotatory motions of the entire masses. He uses the term Centripetation, as descriptive of the local effect, to avoid the ambiguity of the term Attraction, which is used to express a cause, and the term universal Gravitation, used to express a universal cause; which cause is local and particular in each planet, as resulting from its own motions. Centripetation, therefore, according to the new system, is a local effect, producing aggregation in the planeCENTRE of Percussion, in a mov-tary masses, while the mutual ac.ng body, is that point where the tion and re-action of distant plapercussion or stroke is the greatest, nets, and of the sun on the planets, Лu. xx 1 x that is, the centre of oscillation is of the whole length from the point of suspension. If the centre of oscillation be made the point of suspension, the point of suspension will become the centre of oscillation. |