THE INCLINED PLANE WITHOUT FRICTION.

258. The mechanical powers now to be considered are often used for other purposes beside those of raising great weights. For example: the parts of a structure have to be forcibly drawn together, a powerful compression has to be exerted, a mass of timber or other material has to be riven asunder by splitting. For purposes of this kind the inclined plane in its various forms, and the screw, are of the greatest use. The screw also, in the form of the screw-jack, is sometimes used in raising weights. It is principally convenient when the weight is enormously great, and the distance through which it has to be raised comparatively small.

259. We shall commence with the study of the inclined plane. The apparatus used is shown in Fig. 41. a b is a plate of glass 4' long, mounted on a frame and turning round a hinge at a; b d is a circular arc, with its centre at a, by which the glass may be supported; d c is a vertical rod, to which the pulley c is clamped. This pulley can be moved up and down, to be accommodated to the position of a b; the pulley is made of brass, and turns very freely. A little truck r is adapted to run on the plane of glass. The truck is laden to weigh 1 lb., and this weight is unaltered throughout the experiments; the wheels are very free, so that the truck runs with but little friction.

260. But the friction, though small, is appreciable, and it will be necessary to measure the amount and then endeavour to counteract its effect upon the motion. The silk cord attached to the truck is very fine, and its weight is neglected. A series of weights is provided; they are made from pieces of brass wire, and weigh 0·1 lb. and 0·01 lb.: these can easily be hooked into the loop on the cord at p. We first make the plane a b horizontal, and bring down the pulley c so that the cord shall be parallel to the plane; we find that a force must be applied by the cord in order to draw the truck along the plane: this force is of course the friction, and by a suitable weight at p the friction may be said to be counterbalanced. But we cannot expect that the friction will be the same when the plane is horizontal as when the plane is inclined. We must therefore examine this question by a method analogous to that used in Art. 207.

261. Let the plane be elevated until b e, the elevation of b above a d, is 20"; let c be properly adjusted: it is found that when p is O·45 lb. r is just pulled up; and on the other hand, when p is only 0·40 lb. the truck descends and raises p; and when p has any value intermediate between these two, the truck remains in equilibrium. Let us denote the force of gravity acting down the plane by r, and it follows that r must be 0·425 lb., and the friction 0·025 lb. For when p raises r, it must overcome friction as well as r; therefore the power must be 0·025+0·425=0·45. On the other hand, when r raises p, it must also overcome the friction 0·025, therefore p can only be 0·425-0·025=0·40; and r is thus found to be a mean between the greatest and least values of p consistent with equilibrium. If the plane be raised so that the height b e is 33", the greatest and least values of p are 0·66 and 0·71; therefore r is 0·685 friction 0·025, the same as before. Finally, making the height b e only 2", the friction is found to be 0·020, which is not much less than the previous determinations. These experiments show that we may consider this very small friction to be practically constant at these inclinations. (Were the friction large, other methods are necessary, see Art. 265.) As in the experiments r is always raised we shall give p the permanent load of 0·025 lb., thus sufficiently counteracting friction, which we may therefore dismiss from consideration. It is hardly necessary to remark that, in afterwards recording the weights placed at p, this counterpoise is not to be included.

262. We have now the means of studying the relation between the power and the load in the frictionless inclined plane. The incline being set at different elevations, we shall observe the force necessary to draw up the constant load of 1 lb. Our course will be guided by first making use of the principle of energy. Suppose b e to be 2'; when the truck has been moved from the bottom of the plane to the top, it will have been raised vertically through a height of 2', and two units of energy must have been consumed. But the plane being 4' long, the force which draws up the truck need only be 0·5 lb., for 0·5 lb. acting over 4' produces two units of work. In general, if l be the length of the plane and h its height, R the load, and P the power, the number of units of energy necessary to raise the load is R h, and the number of units expended in pulling it up the plane is Pl: hence Rh=Pl, and consequently P:h::R:l; that is, the power is to the height of the plane as the load is to its length. In the present case R=1 lb., l=48"; therefore P=0·0208h, where h is the height of the plane in inches, and P the power in pounds.

263. We compare the powers calculated by this formula with the actual observed values: the result is given in Table XIII.

Table XIII.—Inclined Plane.

Thus for example, in experiment 6, where the height b e is 15", it is observed that the power necessary to draw the truck is 0·31 lb. The truck is placed in the middle of the plane, and the power is adjusted so as to be sufficient to draw the truck to the top with certainty; the necessary power calculated by the formula is also 0·31 lbs., so that the theory is verified.

264. The fifth column of the table shows the difference between the observed and the calculated powers. The very slight differences, in no case exceeding the fiftieth part of a pound, may be referred to the inevitable errors of experiment.

THE INCLINED PLANE WITH FRICTION.

265. The friction of the truck upon the glass plate is always very small, and is shown to have but little variation at those inclinations of the plane which we used. But when the friction is large, we shall not be justified in neglecting its changes at different elevations, and we must adopt more rigorous methods. For this inquiry we shall use the pine plank and slide already described in Art. 117. We do not in this case attempt to diminish friction by the aid of wheels, and consequently it will be of considerable amount.

266. In another respect the experiments of Table XIII. are also in contrast with those now to be described. In the former the load was constant, while the elevation was changed. In the latter the elevation remains constant while a succession of different loads are tried. We shall find in this inquiry also that when the proper allowance has been made for friction, the theoretical law connecting the power and the load is fully verified.

267. The apparatus used is shown in Fig. 33; the plane, is, however, secured at one inclination, and the pulley c shown in Fig. 32 is adjusted to the apparatus, so that the rope from the pulley to the slide is parallel to the incline. The elevation of the plane in the position adopted is 17°·2, so that its length, base, and height are in the proportions of the numbers 1, 0·955, and 0·296. Weights ranging from 7 lbs. to 56 lbs. are placed upon the slide, and the power is found which, when the slide is started by the screw, will draw it steadily up the plane. The requisite power consists of two parts, that which is necessary to overcome gravity acting down the plane, and that which is necessary to overcome friction.

268. The forces are shown in Fig. 42. r g, the force of gravity, is resolved into r l and r m; r l is evidently the component acting down the plane, and r m the pressure against the plane; the triangle g l r is similar to a b c, hence if r be the load, the force r l acting down the plane must be 0·296 r, and the pressure upon the plane 0·955 r.

269. We shall first make a calculation with the ordinary law that the friction is proportional to the pressure. The pressure upon the plane a b, to which the friction is proportional, is not the weight of the load. The pressure is that component (r m) of the load which is perpendicular to the plane a b. When the weights do not extend beyond 56 lbs., the best value for the coefficient of friction is 0·288 (Art. 141): hence the amount of friction upon the plane is

0·288×0·955R=0·275R.

This force must be overcome in addition to 0·296 R (the component of gravity acting down the plane): hence the expression for the power is

0·275R+0·296R=0·571R. 270. The values of the observed powers compared with the powers calculated from the expression 0·571 R are shown in Table XIV.

Table XIV.—Inclined Plane.

271. Thus for example, in experiment 6, a load of 42 lbs. was raised by a force of 24·2 lbs., while the calculated value is 24·0 lbs.; the difference, 0·2 lbs., is shown in the last column.

272. The calculated values are found to agree tolerably well with the observed values, but the presence of the large differences in No. 1 and No. 4 leads us to inquire whether by employing the more accurate law of friction (Art. 141) a better result may not be obtained.

In Table VI. we have shown that the friction for weights not exceeding 56 lbs. is expressed by the formula F=0·9+0·266×pressure, but the pressure is in this case = 0·955 R, and hence the friction is

0·9+0·254R.

To this must be added 0·296R, the component of the force of gravity which must be overcome, and hence the total force necessary is

0·9+0·55R.

The powers calculated from this expression are compared with those actually observed in Table XV.

Table XV.—Inclined Plane.

For example: in experiment 5, a load of 35 lbs. is found to be raised by a power of 20·0 lbs., while the calculated power is 0·9+0·55×35=20·1 lbs.

273. The calculated values of the powers are shown by this table to agree extremely well with the observed values, the greatest difference being only O·3 lb. Hence there can be no doubt that the principles on which the formula has been calculated are correct. This table may therefore be regarded as verifying both the law of friction, and the rule laid down for the relation between the power and the load in the inclined plane.

274. The inclined plane is properly styled a mechanical power. For let the weight be 30 lbs., we calculate by the formula that 17·4 lbs. would be sufficient to raise it, so that, notwithstanding the loss by friction, we have here a smaller force overcoming a larger one, which is the essential feature of a mechanical power. The mechanical efficiency is 30÷17·4=1·72.

275. The velocity ratio in the inclined plane is the ratio of the distance through which the power moves to the height through which the weight is raised, that is 1 ÷ 0·296 = 3·38. To raise 30 lbs. one foot, a force of 17·4 lbs. must therefore be exerted through 3·38 feet. The number of units of work expended is thus 17·4×3·38=58·8. Of this 30 units, equivalent to 51 per cent., are utilized. The remaining 28·8 units, or 49 per cent., are absorbed by friction.

276. We have pointed out in Art. 222 that a machine in which less than half the energy is lost by friction will permit the load to run down when free: this is the case in the present instance; hence the weight will run down the plane unless specially restrained. That it should do so agrees with Art. 147, for it was there shown that at about 13°·4, and still more at any greater inclination, the slide would descend when started.

THE SCREW.

277. The inclined plane as a mechanical power is often used in the form of a wedge or in the still more disguised form of a screw. A wedge is an inclined plane which is forced under the load; it is usually moved by means of a hammer, so that the efficiency of the wedge is augmented by the dynamical effect of a blow.

278. The screw is one of the most useful mechanical powers which we possess. Its form may be traced by wrapping a wedge-shaped piece of paper around a cylinder, and then cutting a groove in the cylinder along the spiral line indicated by the margin of the paper. Such a groove is a screw. In order that the screw may be used it must revolve in a nut which is made from a hollow cylinder, the internal diameter of which is equal to that of the cylinder from which the screw is cut. The nut contains a spiral ridge, which fits into the corresponding thread in the screw; when the nut is turned round, it moves backwards or forwards according to the direction of the rotation. Large screws of the better class, such as those upon which we shall first make experiments, are always turned in a lathe, and are thus formed with extreme accuracy. Small screws are made in a simpler manner by means of dies and other contrivances.

279. A characteristic feature of a screw is the inclination of the thread to the axis. This is most conveniently described by the number of complete turns which the thread makes in a specified length of the screw, usually an inch. For example: a screw is said to have ten threads to the inch when it requires 10 revolutions of the nut in order to move it one inch. The shape of the thread itself varies with the purposes for which the screw is intended; the section may be square or triangular, or, as is generally the case in small screws, of a rounded form.

280. There is so much friction in the screw that experiments are necessary for the determination of the law connecting the power and the load.

281. We shall commence with an examination of the screw by the apparatus shown in Fig. 43.

The nut a is secured upon a stout frame; to the end of the screw hooks are attached, in order to receive the load, which in this apparatus does not exceed 224 lbs.; at the top of the screw is an arm e by which the screw is turned; to the end of the arm a rope is attached, which passing over a pulley d, carries a hook for receiving the power c.

282. We first apply the principle of work to this screw, and calculate the relation between the power and the load as it would be found if friction were absent. The diameter of the circle described by the end of the arm is 20"·5; its circumference is therefore 64"·4. The screw contains three threads in the inch, hence in order to raise the load 1" the power moves 3×64"·4=193" very nearly; therefore the velocity ratio is 193, and were the screw capable of working without friction, 193 would represent the mechanical efficiency. In actually performing the experiments the arm E is placed at right angles to the rope leading to the pulley, and the power hook is weighted until, with a slight start, the arm is steadily drawn; but the power will only move the arm a few inches, for when the cord ceases to be perpendicular to the arm the power acts with diminished efficiency; consequently the load is only raised in each experiment through a small fraction of an inch, but quite sufficient for our purpose.

Table XVI.—The Screw.

283. The results of the experiments are shown in Table XVI. If the motion had not been aided by a start the powers would have been greater. Thus in experiment 6, 2·4 lbs. is the power with a start, when without a start 3·2 lbs. was found to be necessary. The experiments have all been aided by a start, and the results recorded have been corrected for the friction of the pulley over which the rope passes: this correction is very small, in no case exceeding 0·2 lb. The fourth column contains the values of the powers computed by the formula P=0·0143R. This formula has been deduced from the observations in the manner described in the Appendix. The fifth column proves that the experiments are truly represented by the formula: in each of the experiments 7 and 8, the difference between the calculated and observed values amounts to 0·1 lb., but this is quite inconsiderable in comparison with the weights we are employing.

284. In order to lift 100 lbs. the expression 0·0143R shows that 1·43 lbs. would be necessary: hence the mechanical efficiency of the screw is 100÷1·43=70. Thus this screw is vastly more powerful than any of the pulley systems which we have discussed. A machine so capable, so compact, and so strong as the screw, is invaluable for innumerable purposes in the Arts, as well as in multitudes of appliances in daily use.

285. It is evident, however, that the distance through which the screw can raise a weight must be limited by the length of the screw itself, and that in the length of lift the screw cannot compete with many of the other contrivances used in raising weights.

286. We have seen that the velocity ratio is 193; therefore, to raise 100 lbs. 1 foot, we find that 1·43×193=276 units of energy must be expended: of this only 100 units, or 36 per cent., is usefully employed; the rest being consumed in overcoming the friction of the screw. Thus nearly two-thirds of the energy applied to such a screw is wasted. Hence we find that friction does not permit the load to run down, since less than fifty per cent. of the applied energy is usefully employed (Art. 222). This is one of the valuable properties which the screw possesses.

287. We may contrast the screw with the pulley-block (Art. 199). They are both powerful machines: the latter is bulky and economical of power, the former is compact and wasteful of power; the latter is adapted for raising weights through considerable distances, and the former for exerting pressures through short distances.

THE SCREW-JACK.

288. The importance of the screw as a mechanical power justifies us in examining another of its useful forms, the screw-jack. This machine is used for exerting great pressures, such for example as starting a ship which is reluctant to be launched, or replacing a locomotive upon the line from which its wheels have slipped. These machines vary in form, as well as in the weights for which they are adapted; one of them is shown at d in Fig. 44, and a description of its details is given in Table XVII. We shall determine the powers to be applied to this machine for overcoming resistances not exceeding half a ton.

289. To employ weights so large as half a ton would be inconvenient if not actually impossible in the lecture room, but the required pressures can be produced by means of a lever. In Fig. 44 is shown a stout wooden bar 16' long. It is prevented from bending by means of a chain; at e the lever is attached to a hinge, about which it turns freely; at a a tray is placed for the purpose of receiving weights. The screw-jack is 2' distant from e, consequently the bar is a lever of the second order, and any weight placed in the tray exerts a pressure eightfold greater upon the top of the screw-jack. Thus each stone in the tray produces a pressure of 1 cwt. at the point d. The weight of the lever and the tray is counterpoised by the weight c, so that until the tray receives a load there is no pressure upon the top of the screw-jack, and thus we may omit the lever itself from consideration. The screw-jack is furnished with an arm d g; at the extremity g of this arm a rope is attached, which passes over a pulley and supports the power b.

290. The velocity ratio for this screw-jack with an arm of 33", is found to be 414, by the method already described (Art. 283). 291. To determine its mechanical efficiency we must resort to experiment. The result is given in Table XVII.

Table XVII.—The Screw-Jack.

292. It may be seen from the column of differences how closely the experiments are represented by the formula. The power which is required to raise a given weight, say 600 lbs., may be calculated by this formula; it is 0·66+0·0075×600=5·16. Hence the mechanical efficiency of the screw-jack is 600 ÷ 5·16 = 116. Thus the screw is very powerful, increasing the force applied to it more than a hundredfold. In order to raise 600 lbs. one foot, a quantity of work represented by 5·16×414=2136 units must be expended; of this only 600, or 28 per cent., is utilized, so that nearly three-quarters of the energy applied is expended upon friction.

293. This screw does not let the load run down, since less than 50 per cent. of energy is utilised; to lower the weight the lever has actually to be pressed backwards.

294. The details of an experiment on this subject will be instructive, and afford a confirmation of the principles laid down. In experiment 10 we find that 9·0 lbs. suffice to raise 1,120 lbs.; now by moving the pulley to the other side of the lever, and placing the rope perpendicularly to the lever, I find that to produce motion the other way—that is, of course to lower the screw—a force of 3·4 lbs. must be applied. Hence, even with the assistance of the load, a force of 3·4 lbs. is necessary to overcome friction. This will enable us to determine the amount of friction in the same manner as we determined the friction in the pulley-block (Art. 207). Let x be the force usefully employed in raising, and y the force of friction, which acts equally in either direction against the production of motion; then to raise the load the power applied must be sufficient to overcome both x and y, and therefore we have x+y=9·0. When the weight is to be lowered the force x of course aids in the lowering, but x alone is not sufficient to overcome the friction; it requires the addition of 3·4 lbs., and we have therefore x+3·4=y, and hence x=2·8, y=6·2.

That is, 2·8 is the amount of force which with a frictionless screw would have been sufficient to raise half a ton. But in the frictionless screw the power is found by dividing the load by the velocity ratio. In this case 1120÷414=2·7, which is within 0·1 lb. of the value of x. The agreement of these results is satisfactory.

THE SCREW BOLT AND NUT.

295. One of the most useful applications of the screw is met with in the common bolt and nut, shown in Fig. 45. It consists of a wrought iron rod with a head at one end and a screw on the other, upon which the nut works. Bolts in many different sizes and forms represent the stitches by which machines and frames are most readily united. There are several reasons why the bolt is so convenient. It draws the parts into close contact with tremendous force; it is itself so strong that the parts united practically form one piece. It can be adjusted quickly, and removed as readily. The same bolt by the use of washers can be applied to pieces of very different sizes. No skilled hand is required to use the simple tool that turns the nut. Adding to this that bolts are cheap and durable, we shall easily understand why they are so extensively used.

296. We must remark in conclusion that the bolt owes its utility to friction; screws of this kind do not overhaul, hence when the nut is screwed home it does not recoil. If it were not that more than half the power applied to a screw is consumed in friction, the bolt and the nut would either be rendered useless, or at least would require to be furnished with some complicated apparatus for preventing the motion of the nut.