APPLIED MECHANICS] MECHANICS 7G5 other. Now from the laws of statics (see above) it is known that, iu onler that a system offerees applied to a system of connected points may be in equilibrium, it is necessary that the sum formed by putting together the products of the forces by the respective distances through which their points of application are capable of moving simultaneously, each along the direction of the force applied to it, shall be zero, products being considered positive or negative according as the direction of the forces and the possible motions of their points of application are the same or opposite. In other words, the sum of the negative products is equal to the sum of the positive products. This principle, applied to a machine whose parts move with uniform velocities, is equivalent to saying that in any given interval of time the energy exerted is equal to the work performed. The symbolical expression of this law is as follows : let efforts be applied to one or any number of points of a machine ; let any one of these efforts be represented by P, and the distance traversed by its point of application in a given interval of time by ds ; let resistances be overcome at one or any number of points of the same machine ; let any one of these resistances be denoted by R, and the distance traversed by its point of application in the given interval of time by ds ; then ...... (52). The lengths ds, ds are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I. 100. Efficiency. The efficiency of a machine is the ratio of the useful work to the total work, that is, to the energy exerted, and is represented by Z.Rvds ^ __ 2 ?."*/ _ = 2.~R u ds = U ,^ ~?TRds ~ 27R^ds r ~+2.R p ds ~ Z.l>ds~ E K, t being taken to represent useful and R p prejudicial resistances. The more nearly the efficiency of a machine approaches to unity the better is the machine. 101. Power and Effect. The power of a machine is the energy exerted, and the effect the useful work performed, in some interval of time of definite length, such as a second, an hour, or a day. The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour. 102. Modulus of a Machine. In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances are generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement ; and, having been determined, they serve for the computation of the lost work, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the modulus of the machine. The general form of the modulus may be expressed thus ..... (54), where A denotes some quantity or set of quantities depending on the form, arrangement, weight, and other properties of the mechanism. Moseley, however, has pointed out that iu most cases this equation takes the much more simple form of E = (H-A)U-fB ....... (55), where A and B are constants, depending on the form, arrangement, and weight of the mechanism. The efficiency corresponding to the last equation is U 1 f , P . E-1+A + B/U ....... 103. Trains of Mechanism. In applying the preceding prin ciples to a train of mechanism, it may either be treated as a whole, or it may be considered in sections consisting of single pieces, or of any convenient portion of the train, each section being treated as a machine driven by the effort applied to it and energy exerted upon it through its line of connexion with the preceding section, performing useful work by driving the following section, and losing work by overcoming its own prejudicial resistances. It is evident that the efficiency of the whole train is the product of tlie efficiencies of its sections. 104. Rotating Pieces Couples of Forces. It is often convenient to express the energy exerted upon and the work performed by a turning piece in a machine in terms of the moment of the couples of forces acting on it, and of the angular velocity. See p. 728, 219. The ordinary British unit of moment is a foot-pound ; but it is to be remembered that this is a foot-pound of a different sort from the unit of energy and work. If a force be applied to a turning piece in a line not passing through its axis, the axis will press against its bearings with an equal and parallel force, and the equal and opposite reaction of the bearings will constitute, together with the first-mentioned force, a couple whose arm is the perpendicular distance from the axis to the line of action of the first force. A couple is said to be right or left handed with reference to the observer, according to the direction in which it tends to turn the body, and is a driving couple or a resisting couple according as its tendency is with or against that of the actual rotation. Let dt be an interval of time, o the angular velocity of the piece ; then adt is the angle through which it turns in the interval dt, and ds = vdt = rod t is the distance through which the point of application of the force moves. Let P represent an effort, so that IV is a driving couple, then , Yds Yvdt = Tradt = Marf< (57) is the energy exerted by the couple M in the interval dt ; and a similar equation gives the work performed in overcoming a resisting couple. When several couples act on one piece, the resultant of their moments is to be multiplied by the common angular velocity of the whole piece. 105. Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece. In computations respecting machines it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, the equivalent force or couple applied to some other point or piece ; that is to say, the force or couple, which, if applied to the other point or piece, would exert equal energy or employ equal work. The principles of this reduc tion are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of applica tion, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied. These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed. 106. Balanced Lateral Pressure of Guides and Bearings. The most important part of the lateral pressure on a piece of mechanism is; the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece ; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included. There may be or may not be an unbalanced component in this pressure, due to the deviated motion. Its laws will be considered in the sequel. 107. Friction Unguents. The most important kind of resistance in machiues is the friction or rubbing resistance of surfaces which slide over each other. The direction of the resistance of friction is opposite to that in which the sliding takes place. Its magnitude is the product of the normal pressure or force which presses the rubbing surfaces together in a direction perpendicular to themselves into a specific constant already mentioned in Part I., sect. 13, as the coefficient of friction, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered. The total pressure exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity, or inclination to the common perpendicular of the surfaces, is the angle of repose formerly mentioned in sect. 13, whose tangent is the coefficient of friction. Thus, let N be the normal pressure, R the friction, T the total pressure, / the co efficient of friction, and <j> the angle of repose ; then . . (58). R=/N = Ntan0 = Tsin0 ) Experiments on friction have been made by Coulomb, Vince, Rennie, Wood, D. Rankine, and others. The most complete and elaborate experiments are those of Morin, published in his Notions Fondamentalcs de Mecanique, and republished in Britain in tho works of Moseley and Gordon. The following is an exceedingly condensed abstract of the most important results, as regards machines, of these experiments : Surfaces. / Wood on wood, dry 2 Vo 5 Do., soaped r 2 Motals on oak, dry n24 to" 0-26 Do., wet 024 n .o Do., soaped not fr9B Do., elm, dry ({."a* Hemp on oak, dry " Do., wet n-27t e-"S Leather on oak, wet or dry u (..? f " Leather on metals, dry Po., wet Do.^ preasy 23 Do., Oiled ft-IBtn 0-2 Metals on metals, dry u l .J u *
Smooth surfaces with un gents , occaM()na^y greased . O-OTJo^O-OS