OS of the inclined plane to its base OH. In the first case the pressure of the solid on t plane is expressed by OH, and in the second by SH: hence we have In the first case it must be observed, that the effect of the force being parallel to the in elined plane, it neither increases nor diminishes the pressure upon that plane; and this the most favourable case for keeping a body in equilibrio on an inclined plane. In th second case, the direction forming an acute angle with the plane uselessly augments th load or weight. Whilst the direction of the force forms an obtuse angle with the in clination of the plane, by sustaining a portion of the weight, it diminishes the load on th plane, but requires a greater force. 1302. The force necessary to sustain upon an inclined plane a body whose base 1 formed by a plane surface depends, as we have already observed, on the roughness of th surfaces, as well of the inclined plane as of the base of the body; and it is only to be dis covered by experiment. 1303. Of all the means that have been employed to estimate the value of the resistance known under the name of friction, the simplest, and that which seems to give the truesi results, is to consider the inclination of the plane upon which a body, the direction of whose centre of gravity does not fall out of the base, remains in equilibrio, as a horizontal plane; after which the degrees of inclination may begin to be reckoned, by which we find that a body which does not begin to slide till the plane's inclination exceeds 30 degrees, being placed on an inclined plane of 45, will not require a greater force to sustain it than a convex body of the same weight on an inclined plane of 15 degrees. 1304. All that has been said on the force necessary to retain a body upon an inclined plane, is applicable to solids supported by two planes, considering that the second acts as a force to counterpoise the first, in a direction perpendicular to the second plane. 1305. When the directions of three forces, PG, QG, GR, meet in the same point G (fig. 551.), it follows, from the preceding observations on the parallelogram of forces, that to be in equilibrium their proportion will be expressed by the three sides of a triangle formed by perpendiculars to their directions; whence it follows, that if through the centre of gravity G of a solid, supported by two planes or by some other point of its vertical direction, lines be drawn perpendicular to the directions of the forces, if equilibrium exist, so will the following proportion, viz. P: Q: R::BA: BC : AC. A Fig. 551. 1306. Lastly, considering that in all sorts of triangles the sides will between each other be as the sines of their opposite angles, we shall have P Q R:: sin. BCA: sin. BAC: sin. ABC; and as the angle BCA is equal to the angle CAD, and CBA to BAE, we shall have P: Q: R:: sin. CAD: sin. BAC sin. BAE; that is, that the weight is represented by the sine of the angle formed by the two inclined planes, and that the pressures upon each of these planes are reciprocally proportional to the sines of the angles which they form with the horizon. THE WHEEL AND AXLE. B D P 1307. The wheel and axle, sometimes called the axis in peritrochio, is a machine consisting of a cylinder C and a wheel B (fig. 552.) having the same axis, at the two extremities of which are pivots on which the wheel turns. The power is applied at the circumference of the wheel, generally in the direction of a tangent by means of a cord wrapped about the cylinder in order to overcome the resistance or elevate the weight. Here the cord by which the power P acts is applied at the circumference of the wheel, while that of the weight W is applied round the axle or another small wheel attached to the larger, and having the same axis or centre C. Thus BA is a lever moveable about the point C, the power P always acting at the distance BC, and the weight W at the distance CA. Therefore P: W:: CA: CB. That is, the weight and power will be in equilibrio when the power P is to the weight W reciprocally as the radii of the circles where they act, or as the radius of the axle CA, where the weight hangs, to the radius of the wheel CB, where the power acts; or, as before, P: W:: CA: CB. P Fig. 552. W 1308. If the wheel be put in motion, the spaces moved through being as the circum ferences, or as the radii, the velocity of W will be to the velocity of P as CA to CB; that is the weight is moved as much slower as it is heavier than the power. Hence, what is gained in power is lost in time; a property common to machines and engines of every class. 1909. If the power do not act at right angles to the radius CB, but obliquely, draw CD perpendicular to the direction of the power, then, from the nature of the lever, P: W::CA: CD. B 1310. It is to the mechanical power of the wheel and axle that belong all turning or wheel machines of different radii; thus, in the roller turning on the axis or spindle CE (fig. 553.) by the handle CBD, the power applied at B is to the weight W on the roller, as the radius of the roller is to the radius CB of the handle. The same rule applies to all cranes, capstans, windlasses, &c.; the power always being to the weight as is the radius or lever at which the weight acts to that at which the power acts; so that they are always in the reciprocal ratio of their velocities. To the same principle are referable the gimlet and auger for boring holes. Fig. 553. 1311. The above observations imply that the cords sustaining the weights are of no sensible thickness. If they are of considerable thickness, or if there be several folds of them over one another on the roller or barrel, we must measure to the middle of the outermost rope for the radius of the roller, or to the radius of the roller must be added half the thickness of the cord where there is but one fold. 1312 The power of the wheel and axle possesses considerable advantages in point of onvenience over the simple lever. A weight can be raised but a little way by a simple lever, whereas by the continued turning of the wheel and axle a weight may be raised to any height and from any depth. 1313. By increasing the number of wheels, moreover, the power may be increased to any extent, making the less always turn greater wheels, by means of what is called tooth and pinion work, wherein the teeth of one circumference work in the rounds or pinions of another to turn the wheel. In case, here, of an equilibrium, the power is to the weight as the continual product of the radii of all the axles to that of all the wheels. So if the power P (fig. 554.) turn the wheel Q, and this turn the small wheel or axle R, and this turn the wheel S, and this turn the axle T, and this turn the wheel V, and this turn the axle X, which raises the weight W; then P: W::CB. DE. FG: AC. BD. EF. And in Fig. 554. the same proportion is the velocity of W slower than that of P. Thus, if each wheel be to its axle as 10 to 1, then P: W::13: 103, or as 1 to 1000. Hence a power of one pound will balance a weight of 1000 pounds; but when put in motion, the power will move 1000 times faster than the weight. 1314. We do not think it necessary to give examples of the different machines for raising weights used in the construction of buildings: they are not many, and will be hereafter named and described. OF THE PULLEY. 1315. A pulley is a small wheel, usually made of wood or brass, turning about a metal axis, and enclosed in a frame, or case, called its block, which admits of a rope to pass freely over the circumference of the pulley, wherein there is usually a concave groove to prevent the rope slipping out of its place. The pulley is said to be fixed or moveable as its block is fixed or rises and falls with the weight. An assemblage of several pulleys is called a system of pulleys, of which some are in a fixed block and the rest in a moveable one. 1316. If a power sustain a weight by means of a fixed pulley, the power and weight are equal. For if through the centre C (fig. 555.) of the pulley we draw the horizontal diameter AB; then will AB represent a lever of the first kind, its prop being the fixed centre C, from which the points A and B, where the power and weight act, being equally distant, the power P is consequently equal to the weight W. 1317. Hence, if the pulley be put in motion, the power P will descend as fast as the weight W ascends: so that the power is not increased by the use of the fixed pulley, even though the rope go over several of them. It is, nevertheless, of great service in the raising of weights, both by changing the direction of the force, for the convenience of acting, and by enabling a person to raise a weight to any height without moving from his place, and also by permitting a great number of persons to exert, at the same time, their force on the rope at P, which they could not do to the weight itself, as is evident in raising the weight, or monkey, as it is called, of a pile-driver, also on many other occasions. Fig. 555. B W 1318. When a pulley is moveable the power necessary to sustain a weight is equal to the half of such weight. For in this case AB (fig. 556.) may be con B WW W Fig. 556. Fig. 557. sidered as a lever of the second kind, the weight being at C, the power acting at A, and the prop or fixed point at B. Then, because P: W::CB: AB and CB= AB, we have P=W or W=2P. 1319. From which it is manifest that when the pulley is put in motion the velocity of the power is double that of the weight, inasmuch as the point P descends twice as fast as the point C and the weight W rises. It is, moreover, evident that the fixed pulley F makes no difference in the point P, but merely changes the motion of it in an opposite direction. 1320. We may hence ascertain the effect of a combination or system of any number of fixed and moveable pulleys, and we shall thereby find that every cord going over a moveable pulley doubles the powers, for each end of the rope bears an equal share of the weight, whilst each rope fixed to a pulley only increases the power by unity. PW, and in fig. 558., P=w=w+w+w 6 OF THE WEDGE. 1321. The wedge is a body in the form of a half rectangular prism, in practice usually of wood or metal. AF or BG (fig. 559.) is the breadth of its back, CE its height, CG, CB its sides, and its end, GBC, is the terminating surface of two equally inclined planes GCE, BCE. 1322. When a wedge is in equilibrio, the power acting on the back is to the force acting at right angles to either side as the breadth of the back AB (fig. 560.) is to the length of the side AC or BC. For three forces which sustain each other in equilibrio are as the corresponding sides of a triangle drawn perpendicular to the directions in which they act. But AB is perpendicular to the force D In fig. 557. W P acting on the back to drive the wedge forward, and the sides AC, BC are perpendicular to the forces acting on them, the three forces are therefore as AB, AC, BC. Thus, the force on the back, its effect perpendicularly to AC, and its effect parallel to AB, are as the three lines AB, AC, and DC, which are perpendicular to them. Hence the thinner the wedge the greater its effect to split any body or to overcome a resistance against the ides of the wedge. 1323. We are, however, to recollect that the resistance or the forces in question are relative to one side only of the wedge; for if those against both sides are to be reckoned, we can take only half the back AD, or else we must take double the line AC or DC. In the wedge the friction is very great, and at least equal to the force to be overcome, inasmuch as it retains any position to which it is driven, whence the resistance is doubled by the friction. But on the other hand, the wedge has considerable advantage over all the other powers, because of the force of the blow with which the back is struck, a force vastly greater than the dead weight or pressure employed in other machines. On this account it is capable of producing effects vastly superior to those of any other power, such as splitting rocks, raising the largest and heaviest bodies by the simple blow of a mallet; objects which could never be accomplished by any simple pressure whereof in practice application could be made. OF THE SCREW. 1324. The screw is a cord wound in a spiral direction round the periphery of a cylinder, and is therefore a species of inclined plane, whose length is to its height as the circumference of the cylinder is to the distance between two consecutive threads of the screw. It is one of the six mechanical powers used in pressing or squeezing bodies close, and is occasionally used in raising weights. 1325. The screw, then, being an inclined plane or half wedge, the force of a power applied in turning it round is to the force with which it presses upwards or downwards, without estimating friction, as the distance between two threads is to the circumference where the power is applied. For considering it as an inclined plane whose height is the distance between two threads, and its base the circumference of the screw; the force in the horizontal direction being to that in the vertical one as the lines perpendicular to them, namely, as the height of the plane or distance between two threads, is to the base of the plane or circumference of the screw; the power, therefore, is to the pressure as the distance of two threads is to the circumference. But in the application of the screw a handle or lever is used, by means whereof the gain in power is increased in the proportion of the radius of the screw to the radius of the power, that is, the length of the handle, or as their circumferences. Consequently the power is to the pressure as the distance of the threads is to the circumference described by the power. The screw being put in motion, the power is then to the weight which would keep it in equilibrio as the velocity of the latter is to that of the former; and hence their momenta are equal, and produced by multiplying each weight or power by its own velocity. 1326. Thus it is a general property of all the mechanical powers, that the momentum of a power is equal to that of the weight which would keep it in equilibrio, or that each of them is proportional to its velocity. 1327. From the foregoing observations, we may be easily led to compute the force exerted by any machine whose action is exerted through the means of the screw. In fig. 561., representing a press driven by a screw whose threads are each one quarter of an inch apart, let it be turned by a handle or lever 4 feet long from A to B. Then supposing the natural force of a man, by which he can lift, pull, or draw, to be 150 pounds, and that it be required to ascertain with what force the screw will press on the board at D when the man turns with his whole force the handle at A and B; we have AB, the diameter of the power, 4 feet or 48 inches; its circumference, therefore, 48 x 3.1416, or 150 nearly; and the distance of ice threads being one quarter of an inch, the power is to the pressure as I to 603. But the power is equal to 150 pounds; therefore, as 1 : 603::150: 90480, and the pressure therefore at D is equal to a weight of 90480 pounds, independent of friction. 1328. In the endless screw AB (fig. 562.), turned by a handle AC of 20 inches radius, the threads of the screw are at a distance of half an inch; and the screw turns a toothed wheel E whose pinion L acts in turning upon another wheel F, and the pinion M of this last wheel acts upon a third wheel G, to the pinion or barrel whereof is hung the weight W. If we would know what weight can be raised through the means of this combination by a man working the handle C, supposing the diameters of the wheels to be 18 inches, and those of the pinions and barrel 2 inches, the teeth and pinions being all similar in size; we Fig. 561. have 20 x 3.1416 x 2-125-664, the circumference of B the power; and 125-664 to 4, or 251-328 to 1, is the force of the screw alone. Again, 18: 2 or 9:1, being the proportion of the wheels to the pinions, and there being three of them, 93: 1 or 729:1 is the power gained by the wheels. 1329. Consequently 251-328 x 729 to 1, or 183218 to 1 nearly, is the ratio of the power to the weight arising from the joint advantage of the screw and the wheels. The power, however, is 150 pounds; therefore 150 x 183218 or 27482716 pounds is the weight the man can sustain, equal to 12269 tons. 1330. It must be observed, that the power has to overcome not only the weight, but at the same time the friction undergone by the screw, which in some cases is so great as to be equal to the weight itself; for it is sometimes sufficient to sustain the weight when the power is taken off. OF FRICTION. Fig. 562. 1331. Though in a preceding page we have slightly touched on the effect of friction, it is to be kept in mind that the foregoing observations and rules have assumed the mechanical powers to be without weight and friction. This is far from the fact; and, however theoretically true all that has hitherto been advanced, very great allowances must be made in practice when power is applied to mechanical purposes, in which a great portion of their effect is lost by friction, inertia, &c. The word friction, properly meaning the act of one body rubbing on another, is in mechanics used to denote the degree of retardation or obstruction to motion which arises from one surface rubbing against another. A heavy body placed upon another is not in a state of equilibrium between all the forces which act upon it, otherwise it could be moved by the application of the smallest force in a direction parallel to the plane. This want of equilibrium results from unbalanced force occasioned by the friction on a level surface. Now if a new force of equal magnitude be applied to counterpoise such unbalanced force, the body will obey the smallest impulse in such direction, and the force thus employed will exactly measure the retarding force of friction. It has been well observed, that friction destroys, but never generates motion; being therein unlike gravity or the other forces, which, though they may retard motion in one direction, always accelerate it in the opposite. Thus the law of friction violates the law of continuity, and cannot be accurately expressed by any geometrical line, nor by any algebraic formula. The author (Playfair, Outlines of Natural Philosophy) just quoted, continues: "Though friction destroys motion and generates none, it is of essential use in mechanics. It is the cause of stability in the structure of machines, and it is necessary to the exertion of the force of animals. A nail or screw or a bolt could give no firmness to the parts of a machine, or of any other structure, without friction. Animals could not walk, or exert their force anyhow, without the support which it affords. Nothing could have any stability, but in the lowest possible situation; and an arch, which could sustain the greatest load when properly distributed, might be thrown down by the weight of a single ounce, if not placed with mathematical exactness at the very point which it ought to occupy." 1332. Many authors have applied themselves to the subject of friction, but the most satis factory results have attended the investigations of the celebrated Coulomb in its application to practical mechanics; and it is to that author we are indebted for the few following succinct observations. I. In the friction of wood upon wood in the direction of the fibres after remaining in contact for one or two minutes, the following mean results were obtained: When oak rubbed upon oak, and the surfaces in contact were reduced to the smallest pos sible dimensions, the friction was 1 1 |