in the 3d column, namely those which put the pulley in motion, we have in the case of a very slow motion the values of the weights which just surmount the stiffness of the cord; these weights are comprised in the 8th column, and differ but little from those calculated immediately and contained in the 9th column. 2x6w 40. Now to know if the greater or less velocity of the weight suspended upon the pulley has any influence upon the resistance due to the stiffness of the cord, we must in the case of the motion calculate what portion of the additional weight hung upon the pulley is employed in overcoming the friction and the rigidity of the cord. Here the formula of a preceding article has its application, w = :: for, the time occupied by the weight in describing the last three feet being nearly the half of that employed in describing the first three feet, the motion may be considered as uniformly accelerated, and the quantities w-w which result, and are contained in the 4th column, differ but little, as is manifest, from the weights employed to overcome the friction and the stiffness of the cords, in the case of an extremely slow motion. And as it appeared from the preceding experiments that the friction was independent of the velocity, or that it opposed the same resistance to the motion, in the different trials for each experiment; it hence follows that the resistance arising from the stiffness of the cord was likewise constant in the same trials, and depended not upon the velocity, at least in any such sensible manner as to merit our regard in computing the powers of machines. 41. The invariableness of the resistance occasioned by the stiffness of cords, under different velocities, appears also immediately from the results comprized in the 5th column of the table, which, as before observed, proves that the motions were nearly uniformly accelerated. And from this property it follows, that there is always a constant part of the weight or power employed in surmounting the friction and stiffness of the cords. "Nevertheless," adds M. Coulomb," it must be acknowledged, that it is not strictly true, that the augmentation of velocity does not augment the resistance due to the rigidity of cordage. This augmentation appears especially perceptible when the cords are stretched with weights or by forces that are under 100 pounds. I have estimated, by many trials, that in such cases a velocity of 8 feet per second would increase by nearly a pound the resistance occasioned by the stiffness of our cord of 30 threads in a yarn: but this augmentation of resistance seems to be a constant quantity for the same degree of velocity, whatever the tension may be: in such sort that it ceases to be perceptible under great tensions, and that there are but very few circumstances in which it may not be neglected in practice: this augmentation with regard to the velocity appears, besides, much greater in new than in old cords, and in tarred cords than in those which are white or untarred." 42. M. Coulomb deduces from these experiments the following general conclusions: (1.) That with respect to practice, in all rotatory machines the ratio of the pressure to the friction may always be supposed constant, and that the influence of the velocity is too small to need our regard. ad (2.) That the resistance which must be overcome to bend a cord over a roller or pulley is represented by a formula composed of two terms; the first is a constant quantity independent of the tension, and of the form (art. 31.) where a is a constant quantity determined by experience, d is a power of the diameter d of the cord, and r the radius of the roller; the second bdn term is 2, where b is a constant quantity, d, n, and r, as before, and a the tension of the cord. Thus the complete for da mula expressing the stiffness of the cord is (a+b). The r power n varies according to the flexibility of the cord, but is usually about 17 or 1'S, or the resistance is nearly proportional to the square of the diameter of the cord: when the cord is much used n decreases to 1.5. or even 14. The following is a summary of results. lbs. 43. Our knowledge of the nature of the friction of axes, and stiffness of cords, though confessedly very imperfect, may be introduced into the computation of the power of machines: this may be illustrated by an example of a capstan or windlass, where the general formula of an equilibrium will be this: where P represents the power, and the other letters as below. The weight to be elevated, is Q = 1000lbs. The radius of the axis or pivot, which is of iron, is This axis turns in a box of copper: the radius of the cylinder about which the cord is rolled, is R' = 10 inches. The arm of the capstan, or the radius, or distance at which the men exert their force, is R10 feet 120 inches. The pivots are supposed to have been plastered with tallow some time, and the instrument often used, till the ratio of the friction to the pressure is reduced to that of experiment 15 in the table of article 35. whence we have that ratio, or ƒ=0·193, and ✔ (1 ++',)= The cord is supposed tarred, and of 120 threads in a yarn, which will support 12 or 14000 lbs. without breaking. Now a tarred cord of 30 threads in a yarn requires a constant effort equivalent to 6.6lbs. to bend it about a roller of 2 inches radius, and an effort proportional to the tension, of 116 lbs. for a quintal, or 116lbs. for 1000 lbs. Here the radius of the cylinder being 10 inches, we must, first supposing the cords equal, diminish these efforts in the ratio of 10 to 2, viz. make their sum =(6.6 +116) for 1000lbs., and (6'6 +8x116) for 8000. And as the cord is of 120 threads in a yarn instead of 30, we must increase the last result, in the ratio of 30 to 120, so shall we have × (6'6 + 928) =747·7 for the effort which will surmount the stiffness of the cord, that is d" (a + b) = 747·7. And since R' 10, we have dn (a + bq) = 7477. These values being substituted in the general formula it be comes P X 120 = (8000 × 10) + 8000 × 2 +7477. or, P = 666'6 + 17·577 + 62·3 = 746.5 lbs. It will be necessary therefore to distribute at the extremities of the bars of the capstan efforts whose sum shall be equivalent to 746-5 lbs.: that is, if a man makes an effort balancing 25 lbs., 30 men will be required to move the weight of 3000ibs. Had there been no friction and were the cords perfectly flexible, the force necessary would have been only or 666-6, less than 8000 12 the other by almost 80 pounds, a difference which is more than equivalent to the force of three men. So that in this example the friction and rigidity of the cord require an increase of between an 8th and a 9th of the whole power which would otherwise have been requisite. This, however, we wish to be received only as an approximation. The details which have been here entered into will, we trust, be found of some utility in directing the practice, and furnish some hints to those who have time and inclination to adopt other series of well-conducted experiments; and thus supply these most important desiderata in practical mechanics. may On the Energy of First Movers. 44. The consideration of the absolute and relative forces of different kinds of first movers is of too great consequence in the application of mechanics to be entirely omitted in this performance: we shall, therefore, present the reader with some observations and tables respecting the chief classes of powers used to drive machinery, viz. water, air, steam, gunpowder, and animal exertion. Water is generally made to operate upon machines by means of its momentum when in motion: but it may also be used, and that as a very powerful mover, when acting by its pressure merely. In the theory of hydrostatics (art. 387.) we explained the principle of the hydrostatical paradox, in which it is asserted that any quantity of water or other fluid may be made to support any other quantity or any weight however great, and indeed to raise the greater weight until it reaches such a height as ensures the equilibrium. Thus in the hydrostatic bellows the weight of a few ounces of water is made to raise several hundred pounds. And in like manner Otto Guericke of Magdeburg made a child balance, and even overcome, the pull exerted by the emperor's six coach horses, merely by sucking the air from beneath a piston. This great power depends upon the fundamental property of fluids, that they press equally in all directions. Mr. Bramah some time ago obtained a patent for a machine acting as a press on this principle of the quaqua versum pressure of fluids: A piston of of an inch diameter forces water into a cylinder of 12 inches diameter, and by this intervention raises the piston of the cylinder: so that a boy acting with a fourth part of his strength on the small piston by means of a lever can raise about 94080 lbs. or 42 tons pressing on the great piston; the increase of power being as 1 to 42 x 12 or 1 to 2304. This contrivance will be more minutely explained under the article BRAMAH's machine, in the alphabetical part of this volume. 45. Asto the effect of water in motion, it will manifestly depend upon the quantity of fluid and its velocity jointly. When the water runs through a notch or an orifice of a regular form situated in the bottom or side of a reservoir, the quantity discharged in any given time may be determined by the rules laid down for those purposes in vol. 1. Book IV. If s2 be the area of any plane exposed to the action of a current of water, and v the velocity per second with which the fluid strikes the plane, then will the force of the fluid be equivalent to the weight of a volume of water where g represents 32 feet, on the suppo expressed by sition that the water strikes the plane directly: but if the fluid strike the plane obliquely and I represent the angle of incidence, the force will be equivalent to the weight of the column 12 sin 21. Or, since a cubic foot of water weighs 624 lbs. 2g 624v s averd. if v and s be expressed in feet we shall have 2g sin 1971502 sin 21 vs2 lbs. averd. for the equivalen weight, which becomes barely 971502v's2 lbs. when the plane is directly opposed to the fluid. See also art. 467. vol, i. 46. In the determination of the velocity of the stream it will be necessary either to ascertain the height h through which the water has fallen freely, as from the end of a spout, when ✔(2gh), or nearly 8h, will shew the velocity, h being in feet; or when the water issues through an orifice in the bottom or side of a reservoir, to have recourse to Chap. 1 and 2. Book IV. vol. I. before referred to. If the stream be ample without much fall such as must necessarily be applied to move an undershot wheel by its impulse, the power will be determinable from the velocity of the water and the quantity which passes through the section of its bed. Dr. Desaguliers, in his Experimental Philosophy, vol. II. pa. 419. gives the following easy method of ascertaining these data: Observe a place where the banks of the river are steep and nearly parallel, so as to make a kind of trough, for the water to run through, and by taking the depth at various places in crossing make a true section of the river. Stretch a string at right angles over it, and at a small distance another parallel to the first. Then take an apple, an orange, or other small ball, just so much lighter than water as to swim in it, and throw it into the water above the strings. Observe when it comes under the first string, by means of a half second pendulum, a stop-watch, or any other proper instrument; and observe likewise when it arrives at the second string. By these means the velocity of the upper surface, which in practice may generally be taken for that of the whole, will be obtained. Aud |