The area of the base of this cylinder is 8A, and its length is yo, consequently the volume generated by SA = OydA; ΘΣ and therefore the whole volume generated = 0Σ (ydA). But if be the distance of the centre of gravity of the y area A from the fixed axis, we have from the nature of the centre of gravity Σ (84) . 7 = Σ (984), or Ay=(ydA); hence the whole volume generated an equation which expresses the property which was to be proved. CHAPTER VII. ON MECHANICAL INSTRUMENTS. 186. EVERY machine, how complicated soever its construction, is found to be reducible to a set of simple ones, called the Mechanical Powers. These, though authors differ considerably on the subject, are generally said to be six in number; viz. These are not the most simple machines; for, rods used in pushing, and cords used in pulling, are much more simple; in fact, every machine will be found to be a combination of levers, cords, and inclined planes, and these might consequently be called the simple Mechanical Powers, with much greater propriety than the six before mentioned. As, however, these are not very complicated in construction and application, and as levers, cords, and inclined planes do always, in actual practice, present themselves in machinery, in one or more of these six combinations, it will very much facilitate our enquiries into any proposed machine, to be acquainted with their forms and the advantages to be expected from their use. In speaking of any machine, the force which is applied to work it is called the working power, or, simply, the Power; the weight to be raised, or resistance to be overcome, is called the Weight; the point where the machine is applied to produce its effect is called the working point; and the fraction Weight is called the Mechanical Advantage (by some authors the Power, but this creates confusion by confounding it with the former definition of power) of the machine. 187. Every machine is useless until put in motion, and therefore its parts ought to be so arranged and adapted that the given power may be able to overcome the proposed weight, and move it with the requisite degree of celerity; but, in discussing the theory of the Mechanical Powers, it will be sufficient to determine the ratio of the weight to the power when they balance each other, for then the slightest addition made to the power will cause it to preponderate and put the machine in motion. 188. It is very important to remark, that when a power is employed in working a machine, a very considerable portion of it is found not to reach the working point, being spent in overcoming the stiffness of the cords and the roughness of surfaces which rub against each other. Much power is also lost through the imperfection of workmanship, the bending of rods, beams and other materials, which are intended to be rigid, the resistance of the air, &c.; but the introduction of the consideration of these things, though very important in a practical point of view, would only tend to embarrass the student by rendering our investigations tedious and perplexing. We shall therefore at first suppose cords to be perfectly flexible, surfaces quite smooth, workmanship geometrically exact, rods and beams perfectly rigid, the air to offer no resistance; &c. "It is scarcely necessary to state, that, all these suppositions being false, none of the consequences deduced from them can be true. Nevertheless, as it is the business of Art to bring machines as near to this state of ideal perfection as possible, the conclusions which are thus obtained, though false in a strict sense, yet deviate from the truth in but a small degree. Like the first outline of a picture, they resemble in their general features that truth, to which, after many subsequent corrections, they must finally approximate. "After a first approximation has been made on the several suppositions which have been mentioned, various effects, which have been previously neglected, are successively taken into account. Roughness, rigidity, imperfect flexibility, the resistance of air and other fluids, the effects of the weight and inertia of the machine, are severally examined, and their laws and properties detected. The modifications and corrections thus suggested, as necessary to be introduced into our former conclusions, are applied, and a second approximation, but still only an approximation to truth is made. For, in investigating the laws which regulate the several effects just mentioned, we are compelled to proceed upon a new group of false suppositions. To determine the laws which regulate the friction of surfaces, it is necessary to assume that every part of the surfaces of contact are uniformly rough; that the solid parts which are imperfectly rigid, and the cords which are imperfectly flexible, are constituted throughout their entire dimensions of a uniform material; so that the imperfection does not prevail more in one part than another. Thus all irregularity is left out of account, and a general average of the effects taken. It is obvious therefore, that by these means we have still failed in obtaining a result exactly conformable to the real state of things; but it is equally obvious, that we have obtained one much more conformable to that state than had been previously accomplished, and sufficiently near it for most practical purposes. "This apparent imperfection in our instruments and powers of investigation, is not peculiar to Mechanics; it pervades all departments of natural science. In Astronomy, the motions of the celestial bodies, and their various changes and appearances, as developed by theory, assisted by observation and experience, are only approximations to the real motions and appearances which take place in nature. It is true that these approximations are susceptible of almost unlimited accu racy; but still they are, and ever will continue to be, only approximations. Optics, and all other branches of natural science, are liable to the same observations *." I. On the Lever. 189. DEF. A Lever is a rigid rod straight or bent, moveable in a certain plane about one of its points, which is fixed and called its fulcrum. 190. In a lever when there is equilibrium the power and weight are to each other inversely as the perpendiculars from the fulcrum upon the directions in which they act. (Both the power and weight are supposed to act in the plane in which the lever is moveable, which is technically called the plane of the lever). Let AB (figs. 40, 42) or AC (fig. 41) or BC (fig. 43), be a lever whose fulcrum is C; A, B the points at which the power P and weight W act; CY, CZ perpendiculars from C upon their directions. Then the equilibrium will not be disturbed by applying at C two forces P', P" parallel and equal to P, and two others W', W" parallel and equal to W. We have thus, six forces acting on the lever, of which (P, P') and (W, W") form two couples, and the two remaining forces P', W' being counterbalanced by the reaction of the fulcrum, may be removed. Hence the couple (P, P'') whose arm is CY, balances the couple (W, W") whose arm is CZ, consequently their moments must be equal; .. P. CYW.CZ 191. To find the pressure on the fulcrum C. We have shewn that P and W are equivalent to two forces P', W' acting at C, and two equal couples (P, P'), (W, W"); these couples may be removed because they are * Captain Kater's Treatise on Machines. |