Part 9

The fourth column shows the calculated values of the powers derived from the formula. It will be seen by the last column that the formula represents the experiments with but little error.

227. Since 60 per cent. of energy is consumed by friction, this machine, like the differential pulley-block, sustains its load when the chains are free. The differential pulley-block gives a mechanical efficiency of 6, while the epicycloidal pulley-block has only a mechanical efficiency of 5, and so far the former machine has the advantage; on the other hand, that the epicycloidal pulley contains but one block, and that its lifting chain has two hooks, are practical conveniences strongly in its favour.

LECTURE VIII. _THE LEVER._ The Lever of the First Order.—The Lever of the Second Order.—The Shears.—The Lever of the Third Order.

THE LEVER OF THE FIRST ORDER.

228. There are many cases in which a machine for overcoming great resistance is necessary where pulleys would be quite inapplicable. To meet these various demands a correspondingly various number of contrivances has been devised. Amongst these the lever in several different forms holds an important place.

229. The lever of the first order will be understood by reference to Fig. 38. It consists of a straight rod, to one end of which the power is applied by means of the weight C. At another point B the load is raised, while at A the rod is supported by what is called the fulcrum. In the case represented in the figure the rod is of iron, 1" × 1" in section and 6' long; it weighs 19 lbs. The power is produced by a 56 lb. weight: the fulcrum consists of a moderately sharp steel edge firmly secured to the framework. The load in this case is replaced by a spring balance H, and the hook of the balance is attached to the frame. The spring is strained by the action of the lever, and the index records the magnitude of the force produced at the short end. This is the lever with which we shall commence our experiments.

230. In examining the relation between the power and the load, the question is a little complicated by the weight of the lever itself (19 lbs.), but we shall be able to evade the difficulty by means similar to those employed on a former occasion (Art. 60); we can counterpoise the weight of the iron bar. This is easily done by applying a hook to the middle of the bar at D, thence carrying a rope over a pulley F, and suspending a weight G of 19 lbs. from its free extremity. Thus the bar is balanced, and we may leave its weight out of consideration.

231. We might also adopt another plan analogous to that of Art. 51, which is however not so convenient. The weight of the bar produces a certain strain upon the spring balance. I may first read off the strain produced by the bar alone, and then apply the weight C and read again. The observed strain is due both to the weight C and to the weight of the bar. If I subtract the known effect of the bar, the remainder is the effect of C. It is, however, less complicated to counterpoise the bar, and then the strains indicated by the balance are entirely due to the power.

232. The lever is 6' long; the point B is 6" from the end, and B C is 5' long. B C is divided into 5 equal portions of 1'; A is at one of these divisions, 1' distant from B, and C is 5' distant from B in the figure; but C is capable of being placed at any position, by simply sliding its ring along the bar.

233. The mode of experimenting is as follows:—The weight is placed on the bar at the position C: a strain is immediately produced upon H; the spring stretches a little, and the bar becomes inclined. It may be noticed that the hook of the spring balance passes through the eye of a wire-strainer, so that by a few turns of the nut upon the strainer the lever can be restored to the horizontal position.

234. The power of 56 lbs. being 4' from the fulcrum, while the load is 1' from the fulcrum, it is found that the strain indicated by the balance is 224 lbs.; that is, four times the amount of the power. If the weight be moved, so as to be 3' from the fulcrum, the strain is observed to be 168 lbs.; and whatever be the distance of the power from the fulcrum, we find that the strain produced is obtained by multiplying the magnitude of the power in pounds by the distance expressed in feet, and fractional parts of a foot. This law may be expressed more generally by stating that _the power is to the load as the distance of the load from the fulcrum is to the distance of the power from the fulcrum_.

235. We can verify this law under varied circumstances. I move the steel edge which forms the fulcrum of the lever until the edge is 2' from B, and secure it in that position. I place the weight C at a distance of 3' from the fulcrum. I now find that the strain on the balance is 84 lbs.; but 84 is to 56 as 3 is to 2, and therefore the law is also verified in this instance.

236. There is another aspect in which we may express the relation between the power and the load. The law in this form is thus stated: _The power multiplied by its distance from the fulcrum is equal to the load multiplied by its distance from the fulcrum_. Thus, in the case we have just considered, the product of 56 and 3 is 168, and this is equal to the product of 84 and 2. The distances from the fulcrum are commonly called the arms of the lever, and the rule is expressed by stating that _The power multiplied into its arm is equal to the load multiplied into its arm_: hence the load may be found by dividing the product of the power and the power arm by the load arm. This simple law gives a very convenient method of calculating the load, when we know the power and the distances of the power and the load from the fulcrum.

237. When the power arm is longer than the load arm, the load is greater than the power; but when the power arm is shorter than the load arm, the power is greater than the load.

We may regard the strain on the balance as a power which supports the weight, just as we regard the weight to be a power producing the strain on the balance. We see, then, that for the lever of the first order to be efficient as a mechanical power it is necessary that the power arm be longer than the load arm.

238. The lever is an extremely simple mechanical power; it has only one moving part. Friction produces but little effect upon it, so that the laws which we have given may be actually applied in practice, without making any allowance for friction. In this we notice a marked difference between the lever and the pulley-blocks already described.

239. In the lever of the first order we find an excellent machine for augmenting power. A power of 14 lbs. can by its means overcome a resistance of a hundredweight, if the power be eight times as far from the fulcrum as the load is from the fulcrum. This principle it is which gives utility to the crowbar. The end of the bar is placed under a heavy stone, which it is required to raise; a support near that end serves as a fulcrum, and then a comparatively small force exerted at the power end will suffice to elevate the stone.

240. The applications of the lever are innumerable. It is used not only for increasing power, but for modifying and transforming it in various ways. The lever is also used in weighing machines, the principles of which will be readily understood, for they are consequences of the law we have explained. Into these various appliances it is not our intention to enter at present; the great majority of them may, when met with, be easily understood by the principle we have laid down.

THE LEVER OF THE SECOND ORDER.

241. In the lever of the second order the power is at one end, the fulcrum at the other end, and the load lies between the two: this lever therefore differs from the lever of the first order, in which the fulcrum lies between the two forces. The relation between the power and the load in the lever of the second order may be studied by the arrangement in Fig. 39.

242. The bar A C is the same rod of iron 72" × 1" × 1" which was used in the former experiment. The fulcrum A is a steel edge on which the bar rests; the power consists of a spring balance H, in the hook of which the end C of the bar rests; the spring balance is sustained by a wire-strainer, by turning the nut of which the bar may be adjusted horizontally. The part of the bar between the fulcrum A and the power C is divided into five portions, each 1' long, and the points A and C are each 6" distant from the extremities of the bar. The load employed is 56 lbs.; through the ring of this weight the bar passes, and thus the bar supports the load. The bar is counterpoised by the weight of 19 lbs. at G, in the manner already explained (Art. 231).

243. The mode of experimenting is as follows:—Let the weight B be placed 1' from the fulcrum; the strain shown by the spring balance is about 11 lbs. If we calculate the value of the power by the rule already given, we should have found the same result. The product of the load by its distance from the fulcrum is 56, the distance of the power from the fulcrum is 5; hence the value of the power should be 56 ÷ 5 = 11·2.

244. If the weight be placed 2' from the fulcrum the strain is about 22·5 lbs. and it is easy to ascertain that this is the same amount as would have been found by the application of the rule. A similar result would have been obtained if the 56 lb. weight had been placed upon any other part of the bar; and hence we may regard the rule proved for the lever of the second order as well as for the lever of the first order: that, the power multiplied by its distance from the fulcrum is equal to the load multiplied by its distance from the fulcrum. In the present case the load is uniformly 56 lbs., while the power by which it is sustained is always less than 56 lbs.

245. The lever of the second order is frequently applied to practical purposes; one of the most instructive of these applications is illustrated in the shears shown in Fig. 40.

The shears consist of two levers of the second order, which by their united action enable a man to exert a greatly increased force, sufficient, for example, to cut with ease a rod of iron 0"·25 square. The mode of action is simple. The first lever A F has a handle at one end F, which is 22" distant from the other end A, where the fulcrum is placed. At a point B on this lever, 1"·8 distant from the fulcrum A, a short link B C is attached; the end of the link C is jointed to a second lever C D; this second lever is 8" long; it forms one edge of the cutting shears, the other edge being fixed to the framework.

246. I place a rod of iron 0"·25 square between the jaws of the shears in the position E, the distance D E being 3"·5, and proceed to cut the iron by applying pressure to the handle. Let us calculate the amount by which the levers increase the power exerted upon F. Suppose for example that I press downwards on the handle with a force of 10 lbs., what is the magnitude of the pressure upon the piece of iron? The effect of each lever is to be calculated separately. We may ascertain the power exerted at B by the rule of moments already explained; the product of the power and its arm is 22 × 10 = 220; this divided by the number of inches, 1·8 in the line A B, gives a quotient 122, and this quotient is the number of pounds pressure which is exerted by means of the link upon the second lever. We proceed in the same manner to find the magnitude of the pressure upon the iron at E. The product of 122 and 8 is 976. This is divided by 3·5, and the quotient found is 279. Hence the exertion of a pressure of 10 lbs. at F produces a pressure of 279 lbs. at E. In round numbers, we may say that the pressure is magnified 28-fold by means of this combination of levers of the second order.

247. A pressure of 10 lbs. is not sufficient to shear across the bar of iron, even though it be magnified to 279 lbs. I therefore suspend weights from F, and gradually increase the load until the bar is cut. I find at the first trial that 112 lbs. is sufficient, and a second trial with the same bar gives 114 lbs.; 113 lbs., the mean between these results, may be considered an adequate force. This is the load on F; the real pressure on the bar is 113 × 27·9 = 3153 lbs.: thus the actual pressure which was necessary to cut the bar amounted to more than a ton.

248. We can calculate from this experiment the amount of force necessary to shear across a bar one square inch in section. We may reasonably suppose that the necessary power is proportional to the section, and therefore the power will bear to 3153 lbs. the proportion which a square of one inch bears to the square of a quarter inch; but this ratio is 16: hence the force is 16 × 3153 lbs., equal to about 22·5 tons.

249. It is noticeable that 22·5 tons is nearly the force which would suffice to tear the bar in sunder by actual tension. We shall subsequently return to the subject of shearing iron in the lecture upon Inertia (Lecture XVI.).

THE LEVER OF THE THIRD ORDER.

250. The lever of the third order may be easily understood from Fig. 39, of which we have already made use. In the lever of the third order the fulcrum is at one end, the load is at the other end, while the power lies between the two. In this case, then, the power is represented by the 56 lb. weight, while the load is indicated by the spring balance. The power always exceeds the load, and consequently this lever is to be used where speed is to be gained instead of power. Thus, for example; when the power, 56 lbs., is 2' distant from the fulcrum, the load indicated by the spring balance is about 23 lbs.

251. The treadle of a grindstone is often a lever of the third order. The fulcrum is at one end, the load is at the other end, and the foot has only to move through a small distance.

252. The principles which have been discussed in Lecture III. with respect to parallel forces explain the laws now laid down for levers of different orders, and will also enable us to express these laws more concisely.

253. A comparison between Figs. 20 and 39 shows that the only difference between the contrivances is that in Fig. 20 we have a spring balance C in the same place as the steel edge A in Fig 39. We may in Fig. 20 regard one spring balance as the power, the other as the fulcrum, and the weight as the load. Nor is there any essential difference between the apparatus of Fig. 38 and that of Fig. 20. In Fig. 38 the bar is pulled down by a force at each end, one a weight, the other a spring balance, while it is supported by the upward pressure of the steel edge. In Fig. 20 the bar is being pulled upwards by a force at each end, and downwards by the weight. The two cases are substantially the same. In each of them we find a bar acted upon by a pair of parallel forces applied at its extremities, and retained in equilibrium by a third force.

254. We may therefore apply to the lever the principles of parallel forces already explained. We showed that two parallel forces acting upon a bar could be compounded into a resultant, applied at a certain point of the bar. We have defined the moment of a force (Art. 64), and proved that the moments of two parallel forces about the point of application of their resultant are equal.

255. In the lever of the first order there are two parallel forces, one at each end; these are compounded into a resultant, and it is necessary that this resultant be applied exactly over the steel edge or fulcrum in order that the bar may be maintained at rest. In the levers of the second and third orders, the power and the load are two parallel forces acting in opposite directions; their resultant, therefore, does not lie between the forces, but is applied on the side of the greater, and at the point where the steel edge supports the bar. In all cases the moment of one of the forces about the fulcrum must be equal to that of the other. From the equality of moments it follows that the product of the power and the distance of the power from the fulcrum equals the product of the load, and the distance of the load from the fulcrum: this principle suffices to demonstrate the rules already given.

256. The laws governing the lever may be deduced from the principle of work; the load, if nearer than the power to the fulcrum, is moved through a smaller distance than the power. Thus, for example, in the lever of the first order: if the load be 12 times as far as the power from the fulcrum, then for every inch the load moves it can be demonstrated that the power must move 12 inches. The number of units of work applied at one end of a machine is equal to the number yielded at the other, always excepting the loss due to friction, which is, however, so small in the lever that we may neglect it. If then a power of 1 lb. be applied to move the power end through 12 inches, one unit of work will have been put into the machine. Hence one unit of work must be done on the load, but the load only moves through ¹/₁₂ of a foot, and therefore a load of 12 lbs. could be overcome: this is the same result as would be given by the rule (Art. 236).

257. To conclude: we have first determined by actual experiment the relation between the power and the load in the lever; we have seen that the law thus obtained harmonizes with the principle of the composition of parallel forces; and, finally, we have shown how the same result can be deduced from the fertile and important principle of work.

LECTURE IX. _THE INCLINED PLANE AND THE SCREW._

The Inclined Plane without Friction.—The Inclined Plane with Friction.—The Screw.—The Screw-jack.—The Bolt and Nut.

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 _P l_: hence _R h_ = _P l_, 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_ = 0·0208 _h_, 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.