PROPOSITION LIII. 270. When a Beam is to sustain any Weight, or Pressure, or Force, acting Laterally; then the Strength ought to be as the Stress upon it; that is, the Breadth multiplied by the Square of the Depth, or in similar sections, the Cube of the Diameter, in every place, ought to be proportional to the Length drawn into the Weight or Force acting on it. And the same is true of several Different Pieces of timber compared together. FOR every several piece of timber or metal, as well as every part of the same, ought to have its strength proportioned to the weight, force, or pressure it is to support. And therefore the strength ought to be universally, or in every part as the stress upon it. But the strength is as the breadth into the square of the depth; and the stress is as the weight or force into the distance it acts at. Therefore these must be in constant ratio. This general property will give rise to the effect of different shapes in beams, according to particular circumstances; as in the following corollaries. 271. Corol. 1. If ABC be a horizontal beam, fixed at the end ac, AC, and sustaining a weight at the other end B. And if the sections at all places be similar figures; and DE be the diameter at any place D; then BD will be every where as DE3. So that if ADB be a right line, then BEC will be a cubic parabola. In which case of such a beam may be cut away, without any diminution of the strength.--But if the beam be bounded by two parallel planes, perpendicular to the horizon; then BD will be as DE2; and then BEC will be the common parabola in which case a 3d part of the beam may be thus cut away. 272. Corol. 2. But if a weight press uniformly on every part of AB; and the sections in all points, as D, be similar; then BD2 will be every where as DE3: and then BEC is the semicubical parabola. But, in this disposition of the weight, if the beam be bounded by parallel planes,perpendicular to the horizon; then BD will be always as DE; and BEC a right line, or ABC a wedge. So that then half the beam may be cut away, without diminution of strength. 273. Corol ༈་ STRENGTH AND STRESS OF BEAMS, &c. 273. Corol. 3. If the beam AB be supported at both ends; and if it sustain a weight at any variable point D, or uniformly on all E : 191 B 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 planes, 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. ་ But if the beam be bounded by two parallel planes, perpendicular to the horizon; then AD is as DE3, 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: Elm, Ash Walnut, Thorn Red fir, Holly, Elder, Plane, Crabtree, Appletree Lead A cylindric rod of good clean fir, of 1 inch circumference, drawn lengthways, will bear at extremity 400 lbs; 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, iron, or a rope of dinches diameter, were to lift of the ex- The fir would bear 84 d2 hundred weights. The rope The iron 22 da ditto. 6a 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 4 times stronger than oak of equal dimensions; and from 5 to 6 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, 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. PROP. PROPOSITION LIV. 280. To find the Centre of Percussion of a Body, or System of Bodies. Let A be the place of one of the particles, so reduced; join sa, and draw AP perpendicular to as, and aa perpendi̟cular to sco; then AP will be the direction of a's motion as it revolves about s; and the whole mass being stopped at o, the body A will urge the point P, 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. (30-SP) -SP) A. sa. so — A . SA A. SA. SPA sa. so A: SA2. In like manner, the forces of в and c, to turn the system about o, are as B But, since the forces on the contrary sides of o destroy one another, by the definition of this force, the sum of the posi tive parts of these quantities must be equal to the sum of the negative parts, that is, A. sa sa. so +B. sb. so+c. sc. so &c. i which is the distance of the centre of percussion below the And here it may be observed that, if any of the points a, b, 281. Corol. 1. Since, by cor. 3, pг. 40, ▲ + в + ¢ &c. percussion, is so = is sc X body b 282. Corol. 2. Since, by Geometry, theor. 36, 37, and SB2 = $G2 + GB2+2SG. Gb, and sc2 SG2 + GC2 + 2SG . GC, &C.; C and, by cor. 5, pr. 40, the sum of the last terms is nothing, OF b. SG2 + A. GA2 + B. or so = sɑ + b. SG GB2 + &c. 283. Corol. 3. Hence the distance of the centre of percussion always exceeds the distance of the centre of gravity, and A. GA2 + B. GB2 &C. the excess is always Go A. GA2 + B. GB2 &c. that is so, Go is always the same constant quantity, where- PROPOSITION 1 |