riable point D, or uniformly on B all parts of its length; and if all the sections be similar figures; then will the diameter DE3 be every where as the rectangle AD DB. But if it be bounded by two parallel plånes, perpendicular to the horizon; then will DE2 be every where as the rectangle AD . DB, and the curve AEB an ellipsis. 274. Corol. 4. But if a weight be placed at any given point F, and all the sections be similar figures; then will AD be as DE3, and AG, BG be two cubic parabolas. A DF E G B But if the beam be bounded by two parallel planes, perpendicular to the horizon; then AD is as De, and AG and BG are two common parabolas. 275. Scholium. The relative strengths of several sorts of wood, and of other bodies, as determined by Mr. Emerson, are as follow: Red fir, Holly, Elder, Plane, Crabtree, Appletree Beech, Cherrytree, Hazle 2 A cylindric rod of good clean fir, of 1 inch circumference, drawn lengthways, will bear at extremity 400lbs; and a spear of fir, 2 inches diameter, will bear about 7 tons in that direction. A rod of good iron, of an inch circumference, will bear a stretch of near 3 tons weight. A good hempen rope, of an inch circumference, will bear 1000 lbs at the most. Hence Mr. Emerson concludes, that if a rod of fir, or of iron, or a rope of d inches diameter, were to lift of the extreme weight; then The fir would bear 84 d2 hundred weights. 22 d2 ditto. 64 d2 tons. Mr. Banks, an ingenious lecturer on mechanics, made many experiments on the strength of wood and metal; whence he concludes, that cast iron is from 3 to 41⁄2 times stronger than oak of equal dimensions; and from 5 to 64 times stronger than deal. And that bars of cast iron, an inch square, weighing 9 lbs. to the yard in length, supported at the extremities, bear on an average, a load of 970 lbs. laterally. And they bend about an inch before they break. Many other experiments on the strength of different materials, and curious results deduced from them, may be seen in Dr. Gregory's and Mr. Emerson's Treatises on Mechanics, as well as some more propositions on the strength and stress of different bars. ON THE CENTRES OF PERCUSSION, OSCILLATION, AND GYRATION. 276. THE CENTRE OF PERCUSSION of a body, or a system of bodies, revolving about a point, or axis, is that point, which striking an immoveable object, the whole mass shall not incline to either side, but rest as it were in equilibrio, without acting on the centre of suspension. 277. The Centre of Oscillation is that point, in a body vibrating by its gravity, in which if any body be placed, or if the whole mass be collected, it will perform its vibrations in the same time, and with the same angular velocity, as the whole body, about the same point or axis of suspension. 278. The 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. 279. The angular motion of a body, or system of bodies, is the motion of a line connecting any point and the centre or axis of motion; and is the same in all parts of the same revolving body. And in different unconnected bodies, each revolving about a centre, the angular velocity is as the absolute velocity directly, and as the distance from the centre inversely; so that, if their absolute velocities be as their radii or distances, the angular velocities will be equal. PRO PROPOSITION LIV. 280. To find the Centre of Percussion of a Body, or System of Bodies. LET the body revolve about an axis passing through any points in the line sGo, passing through the centres of gravity and percussion, G and o. Let MN be the section of the body, or the plane in which the axis SGO moves. And conceive all the particles of the body to be reduced to this plane, by perpendiculars let fall from them to the plane: a supposition which will not affect the centres G, o, nor the angular motion of the body. Let a be the place of one of the particles, so reduced; join sa, and draw AP perpendicular to as, and Aa perpendicular to SGO: then AP will be the direction of A's motion as it revolves about s; and the whole mass being stopped at ó, the body a will urge the point r, forward, with a force proportional to its quantity of matter and velocity, or to its matter and distance from the point of suspensions; that is, as A. SA; and the efficacy of this force in a direction perpendicular to so, at the point p, is as A. sa, by similar triangles; also, the effect of this force on the lever, to turn it about o, being as the length of the lever, is as A. sa PO A. sa. (so - SP) = A. sa SOA. sa. SP = A.Sa.SO A. SA2. In like manner, the forces of в and c, to turn the system about o, are as B. sb so - B. sp2, and But, since the forces on the contrary sides of o destroy one another, by the defi ition of this force, the sum of the positive parts of these quantities must be equal to the sum of the negative parts, that is, A. sa. so + B. sb. so + c.SC so &c = A.SA2+B.SB2 + c. sc2 &c; and 2 A.SA+B SB2 + C. Sc2 &c hence so = A.sa + B.sb + c. sc &c' which is the distance of the centre of percussion below the axis of motion. And here it must be observed that, if any of the points a, b, &c, fall on the contrary side of s, the corresponding product a. sa, or B. sb, &c, must be made negative. 281. Corol. 1. Since, by cor. 3, pr. 40, A + B + C &c, or the body b x the distance of the centre of gravity, sG, is = A sa + B sb + c. sc &c, which is the denominator of the value of so; therefore the distance of the centre of percussion, is so = A.SA+B SB2 + C. Sc2 &c SG × body b 282. Corol. 2. Since, by Geometry, theor. 36, 37, and sB2 = SG2 + GB2 + 2SG. Gb, and sc2 = SG2 + GC2 + 2SG.GC, &c; and, by cor. 5, pr. 40, the sum of the last terms is nothing, namely, 2SG or= Ga + 2SG cb + 2SG.Gc &c = 0; therefore the sum of the others, or A. SA2 + B. SB2 &с is = (A + B &c). SG2 + A. GA2 + B. GB2 + C GC2 &c, b.SG+A. GA2 + B. GB2 + C. GC2 &c; which being substituted in the numerator of the foregoing value of so, gives 283. Corol. 3. Hence the distance of the centre of percussion always exceeds the distance of the centre of gravity, and the excess is always Go 284. And hence also, SG A. GA2 + B. GB2 &C b.SG A. GAB.GB2 &c GO = the body b that is sg. Go is always the same constant quantity, whereever 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 se is greater; and consequently the point o rises upwards and approaches towards the point G, as the points is removed to the greater distance; and they coincide when sg is infinite. But when s coincides with G, then GO is infinite, or o is at an infinite distance. : PROPO PROPOSITION LV. 285. If a Body A, at the Distance sa from an axis passing through s, be made to revolve about that axis by any Force acting at P in the Line SP, Perpendicular to the Axis of Motion: It is required to determine the Quantity or Matter of another Body a, which being placed at P, the Point where the Force acts, it shall be accelerated in the Same Manner, as when A revolved at the Distance SA; and consequently, that the Angular Velocity of a and about s, may be the Same in Both Cases. SP By the nature of the lever, SA : SP::f; -f, the effect of the force f, acting at P, SA on the body at A; that is, the force f acting at P, will have the same effect on the body A, as the force SP SA f, acting directly at the point A. But as ASP revolves altogether about the axis at s, the absolute velocities of the points A and s, or of the bodies a 'and Q, will be as the radii SA, SP, of the circle described by them, Here then we have two bodies A and q, which being urged SP directly by the forces f and saf, acquire velocities which are as sp and SA. And since the motive forces of bodies are as their mass and velocity: therefore SP SA 2 f:f:: A.SA: Q. SP, and SP2: SA2 :: A := و which therefore expresses the mass of matter which, being placed at r, would receive the same angular motion from the action of any force at P, as the body a receives. So that the resistance of any body A, to a force acting at any point p, is directly as the square of its distance sa from the axis of motion, and reciprocally as the square of the distance sp of the point where the force acts. 286. Corol. 1. Hence the force which accelerates the point P, is to the force of gravity, as to A. SA2. f. SP2 A. SA 2 287. Corol. 2. If any number of bodies A, B, C, be put in motion, about a fixed axis passing through s, by a force acting at ; the point p will be accele rated in the same manner, and consequently the whole system will have the same angular velocity, if instead of the 2 to 1, or as f. sp2 |