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. A 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 sidered as a lever of the second the prop or fixed point at B. have P W or W=2P. Fig. 557. kind, the weight being at C, the power acting at A, and Then, because P: W::CB: AB and CB= AB, we 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. In fig. 557. r =W, and in fig. 558., P=w=w+w+w 6 W 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 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 sides 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 the threads being one quarter of an inch, the power is to the pressure as 1 to 603. But the power is equal to 150 pounds; therefore, as 1: 6034::150: 90480, and the pressure therefore at D is equal to a weight of 90480 pounds, independent of friction. Fig. 561. 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 have 20 x 3.1416 x 2=125 664, the circumference of B 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: 1 1383. When the friction was across the grain, or at right angles to the direction of the fibres, oak against oak was 376 The ratios above given are constant quantities, and not dependent upon the velocities, excepting in the case of elm, when the pressures are very small, for then the friction is sensibly increased by the velocity. 1334. (II.) Friction is found to increase with the time of contact. It was ascertained that when wood moved upon wood in the direction of the fibres, the friction gradually increased, and reached its maximum in 8 or 10 seconds. When across the grain of the wood, it took a longer time to reach its maximum. 1335. (III.) For illustration of the friction of metals upon metals after a certain time of rest, the subjoined experiments were made with two flat rulers of iron, 4 feet long and 2 inches wide, attached to the fixed plank of the apparatus used for the investigation. Four other rulers, two of iron and two of brass, 15 inches long and 18 lines wide, were also used. The angles of each of the rulers were rounded off, and the rubbing surfaces of the rulers were 45 square inches. With iron upon iron and a pressure of 53 lbs., the friction in parts of the pressure was 1 36 1 With iron upon brass and a pressure of 52 lbs., the friction in parts of the pressure was 4-2* 452 lbs., 11 1336. In these experiments each set gives nearly the same result, though the second pressures are nearly nine times the first; from which we learn that, in metals, friction is independent of the extent of the rubbing surfaces. Coulomb, moreover, found that the friction is independent of the velocities. The ratio of 4 to 1 between the pressure of friction, in the case of iron moving upon brass, is only to be considered accurate when the surfaces are new and very large. When they are very small the ratio varies from 4 to 1 to 6 to 1; but this last ratio is not reached unless the friction has been continued more than an hour, when the iron and brass have taken the highest polish whereof they are susceptible, free of all scratches. 1337. IV. In the friction of oak upon oak, when greased with tallow, which was renewed at every experiment, some days were required for obtaining, when the surfaces were considerable, the maximum of friction or adhesion. It was nearly similar to that without grease, sometimes rather greater. For iron or copper with tallow, during rest, the increase is not so considerable as with oak. At first the friction was of the weight, besides a small force of a pound for every 30 square inches independent of the weight. The friction after some time changes to or §. Olive oil alters the condition of the friction to, and old soft grease to about. 1338. V. In the case of friction of bodies, oak upon oak for instance, in motion in the direction of its fibres, the friction was nearly constant in all degrees of velocity, though with large surfaces it appeared to increase with the velocities; but when the touching surfaces were very small compared with the pressures, the friction diminished or the velocities increased. For a pressure of 100 to 4000 pounds on a square foot, the friction is about 9.5' besides for each square foot a resistance of 13 pounds, exclusive of pressure increasing a little with the velocity, occasioned perhaps by a down on the surface. If the surface be very small the friction is lessened. When the narrow surface was cross-grained, the friction was invariably. In the case of oak on fir, the friction was; of fir on fir, ; of elm on elm, but varying according to the extent of surface; for iron or copper on wood, which was at first doubled by increasing the velocity to a foot in a second, but on a continuance of the operation for some hours it again diminished. For iron on iron, 3.55; on copper, 4-15; after long attrition, in all velocities. Upon the whole, in the case of most machines, of the pressure may be considered a fair estimate of the friction. 1 5.4 1 1 1 1 1 1339. In the experiments to ascertain the friction of axles, Coulomb used a simple pulley, where the friction of the axis and that of the rigidity of the rope produce a joint resistance. With guaiacum moving upon iron, the friction was or of the weight in all velocities exclusive of the rigidity of the rope; the mean was, or, with a small weight, a little greater. In the cases of axles of iron on copper, or 1 the velocity is small; the friction being always somewhat less than for plane surfaces. With grease, the friction was about With an axis of green oak or elm, and a pulley of guaiacum, the friction with tallow Ws without, with a pulley of elm, the quantities in question became and An axis of box with a pulley of guaiacum gave and; with an elm pulley, and An axis of iron and a pulley of guaiacum gave, with tallow, The velocity had but small 1 7.5 ; 11.5 effect on the rigidity of ropes, except in slightly increasing the resistance when the pressure was small. 1 54.3 1340. The friction and rigidity of ropes was supposed by Amontons and Desaguliers to vary as the diameter as the curvature and as the tension. By Coulomb the power of the diameter expressing the rigidity was found generally to be 17 or 1.8, never less than 14, and that a constant quantity must be supposed as added to the weight. Wet ropes, if small, are more flexible than such as are dry, and tarred ones stiffer by about one sixth, and in cold weather somewhat more. After rest, the stiffness of ropes increases. A rope of three strands, each having two yarns 12 lines in circumference, whose weight was 125 grains, being bent upon an axis 4 inches in diameter, required a constant force of one pound (French) and of the weight to overcome its rigidity. The same rope tarred, required one fifth of a pound and one fiftieth of the weight. When the strands were of fine yarns, the circumference 20 lines, and the weight 347 grains, the rigidity was equal to half a pound and 1 of the weight to move it. With strands of 10 yarns, and a circumference of 28 lines, and a weight of 680 grains to 6 inches, the rigidity of the untarred rope was 2 lbs. and 1 of the weight, and the tarred rope of 3.3 lbs. and of the weight. Experiments which confirmed the above were made on a roller moving on a horizontal plane, while a rope was coiled completely round it, whence an allowance must be made for the friction of the roller on the plane, which varies as its weight and inversely as its diameter. With a roller of guaiacum or lignum vitæ, 3.6 inches in diameter, moving on oak, it was of the weight; for a roller of elm, more. 23.1 1 13.33 10.34 1341. This subject has, we conceive, been pursued as far as is necessary for the architect; seeing that his further investigation of it, should necessity arise, may be accomplished by reference to the works of Amontons, Bulfinger, Parent, Euler, Bossut, and Coulomb, upon whom we have drawn for the information here given. We shall therefore conclude these remarks by subjoining some of the practical results which experiments on animal power afford, extracted from the celebrated Dr. Thomas Young's Natural Philosophy, vol. ii. 1342. In comparing the values of the force of moving powers, it is usual to assume an unit, which is considered as the mean effect of the labour of an active man working to the greatest advantage; this on a moderate calculation will be found sufficient to raise 10 lbs. to the height of 10 feet in one second for 10 hours in a day; or 100 lbs. 1 foot in a second, that is 36,000 feet in a day, or 3,600,000 lbs. 1 foot in a day. The following exhibits a tabular view of the immediate force of men, without deduction for friction. Such a day's work is the measuring unit in the third column of the table. A man weighing 133 lbs. French ascended 62 feet French by steps in 34 seconds, but was completely exhausted. Amontons. A sawyer made 200 strokes of 18 French inches each in 145 seconds, with a force of 25 lbs. French. He could not have continued more than 3 minutes. Amontons. A man can raise 60 French lbs. 1 French foot in A man can exert a force of 40 lbs. for a whole day For a short time, a man may exert a force of 80 lbs. with a fly when the motion is pretty quick. Desaguliers. A man going up stairs ascends 14 metres (35 43 feet) in 1 minute. Coulomb. |