Scholium. The propositions now delivered contain the most material principles in the theory of machines. The manner of applying several of them is very obvious: the application of some, being less manifest, may be briefly illustrated, and the chapter concluded with two or three observations. The last theore.n may be applied to the action of men and of horses, with more accuracy than might at first be supposed. Observations have been made on men and horses drawing a lighter along a canal, and working several days together. The force exerted was measured by the curvature and weight of the track-rope, and afterwards by a spring steelyard. The product of the force thus ascertained, into the velocity per hour, was considered as the momentum. In this way the action of men was found to be very nearly as (w-v): the action of horses loaded so as not to be able to trot was nearly as (w - v)17, or as (w-v). Hence the hypothesis we have adopted may in many cases be safely assumed. According to the best observations, the force of a man at rest is on the average about 70 pounds; and the utmost velocity with which he can walk is about 6 feet per second, taken at a medium. Hence, in our theorems, = 70, and w= 6. Consequently == 313 lbs. the greatest force a man can exert when in motion: and he will then move at the rate of w, or 2 feet per second, or rather less than a mile and a half per hour. The strength of a horse is generally reckoned about 6 times that of a man; that is, nearly 420 lbs. at a dead pull. His utmost walking velocity is about 10 feet per second. Therefore his maximum action will be of 420 = 1863 lbs. and he will then move at the rate of of 10, or 34 feet, per second, or nearly 24 miles per hour. In both these instances we suppose the force to be exerted in drawing a weight along a horizontal plane; or by raising a weight by a cord running over a pulley, which makes its direction horizontal. 2. The theorems just given may serve to show, in what points of view machines ought to be considered, by those who would labour beneficially for their improvement. The first object of the utility of machines consists in furnishing the means of giving to the moving force the most commodious direction; and, when it can be done, of causing its action to be applied immediately to the body to be moved. These can rarely be united: but the former can be accomplished in most instances; of which the use of the simple lever, ever, puiler, and wheel and axle, furnish many examples. The second object gained by the use of machines, is an accuation of the velocity of the work to be performed, to the velocity with which alone a natural power can act. Thus, whenever the natural power acts with a certain velocity which cannot be changed, and the work must be performed with a greater velocity, a machine is interposed moveable round a fixed support, and the distances of the impelled and working points are taken in the proportion of the two given velocities. But the essential advantage of machines, that, in fact, which properly appertains to the theory of mechanics, consists in augmenting, or rather in modifying, the energy of the moving power, in such manner that it may produce effects of which it would have been otherwise incapable. Thus a man might carry up a flight of steps 20 pieces of stone, each weighing 50 pounds (one by one) in as small a time as he could (with the same labour) raise them all together by a piece of machinery, that would have the velocities of the impelled and working points as 20 to 1, and, in this case, the instrument would furnish no real advantage, except that of saving his steps. But if a large block of 20 times 30, or 600 lbs. weight, were to be raised to the same height, it would far surpass the utmost efforts of the man, without the intervention of some such contrivance. The same purpose may be illustrated somewhat differently; confining the attention all along to machines whose motion is uniform. The product fu represents, during the unit of time, the effect which results from the motion of the resistance; this motion being produced in any manner whatever. If it be produced by applying the moving force immediately to the resistance, it is necessary not only that the products FV and fu should be equal; but that at the same time F=ƒ, and v=v: if, therefore, as most frequently happens, ƒ be greater than F, it will be absolutely impossible to put the resistance in motion by applying the moving force immediately to it. Now machines furnish the means of disposing the product FV in such a manner that it may always be equal to fv, however much the factors of FV may differ from the analogous factors in fu; and, consequently, of putting the system in motion, whatever is the excess of ƒ over F. Or, generally, as M. Prony remarks (Archi. Hydraul. art. 504), machines enable us to dispose the factors of Fvt in such a manner, that while that product continues the same, its factors may have to each other any ratio we desire. If, for instance, time be precious, the effect must be produced in a very short short time, and yet we should have at command a force capable of little velocity but of great effort, a machine must be found to supply the velocity necessary for the intensity of the force: if, on the contrary, the mechanist has only a weak power at his disposition, but capable of a great velocity, a machine must be adopted that will compensate, by the velocity the agent can communicate to it, for the force wanted: lastly, if the agent is capable neither of great effort, nor of great velocity, a convenient machine may still enable him to accomplish the effect desired, and make the product Fvt of force, velocity, and time, as great as is requisite. Thus, to give another example: Suppose that a man, exerting his strength immediately on a mass of 25 lbs, can raise it vertically with a velocity of 4 feet per second; the same man acting on a mass of 1000lbs, cannot give it any vertical motion though he exerts his utmost strength, unless he has recourse to some machine. Now he is capable of producing an effect equal to 25 x 4 x t: the letter t being introduced because, if the labour is continued, the value of t will not be indefinite, but comprised within assignable limits. Thus we have 25 × 4 × 1 = 1000 × v x t; and consequently v = of a foot. This man may therefore with a machine, as a lever, or axis in peritrochio, cause a mass of 1000 lbs to rise of a foot, in the same time that he could raise 25 lbs 4 feet without a machine; or he may raise the greater weight as far as the less, by employing 40 times as much time. From what has been said on the extent of the effects which may be attained by machines, it will be seen that, so long as a moving force exercises a determinate effort, with a velocity also determinate, or so long as the product of these is constant, the effect of the machine will remain the same: thus, under this point of view, supposing the preponderance of the effort of the moving power, and abstracting from inertia and friction of materials, the convenience of application, &c, all machines are equally perfect. But, from what has been shown, (props. 9, 10) a moving force may, by diminishing its velocity, augment its effort, and reciprocally. There is therefore a certain effort of the moving force, such that its product by the velocity which comports to that effort, is the greatest possible. Admitting the truth of the law assumed in the propositions just referred to, we have, when the effect is a maximum, v=w, or F = ; and these two values obtaining together, their product w expresses the value of the greatest effect with respect to the unit of time. In practice it will always be adviseable to approach as nearly to these values as circumstances will admit; for it cannot be expected expected that they can always be exactly attained. But a small variation will not be of much consequence: for, by a well-known property of those quantities which admit of a proper maximum and minimum, a value assumed at a moderate distance from either of these extremes will produce no sensible change in the effect. If the relation of F to v followed any other law than that which we have assumed, we should find from the expression of that law values of F, v, &c, different from the preceding. The general method however would be nearly the same. With respect to practice, the grand object in all cases should be to procure a uniform motion, because it is that from which (cæteris paribus) the greatest effect always results. Every irregularity in the motion wastes some of the impelling power; and it is the greatest only of the varying velocities which is equal to that which the machine would acquire if it moved uniformly throughout: for, while the motion accelerates, the impelling force is greater than what balances the resistance at that time opposed to it, and the velocity is less than what the machine would acquire if moving uniformly; and when the machine attains its greatest velocity, it attains it because the power is not then acting against the whole resistance. In both these situations therefore, the performance of the machine is less than if the power and resistance were exactly balanced; in which case it would move uniformly (art. 1). Besides this, when the motion of a machine, and particularly a very ponderous one, is irregular, there are continual repetitions of strains and jolts which soon derange and ultimately destroy the whole structure. Every attention should therefore be paid to the removal of all causes of irregularity. CHAPTER XII. PRESSURE OF EARTH AND FLUIDS AGAINST WALLS AND FORTIFICATIONS, THEORY OF MAGAZINes, &c. PROBLEM 1. To determine the Pressure of Earth against Walls. WHEN new-made earth, such as is used in forming ramparts, &c, is not supported by a wall as a facing, or by counterforts and land-ties, &c, but left to the action of its weight and the weather; the particles loosen and separate from each other, other, and form a sloping surface, nearly regular; which plane surface is called the natural slope of the earth; and is supposed to have always the same inclination or deviation from the perpendicular, in the same kind of soil. In common earth or mould, being a mixture of all sorts thrown together, the natural slope is commonly at about half a right angle, or 45 degrees; but clay and stiff loam stands a greater angle above the horizon, while sand and light mould will only stand at a much less angle. The engineer or builder must therefore adapt his calculations accordingly. Now, we have already given, at prop. 44 Statics in vol. 2, 6th edition, the general theory and determination of the force with which the triangle of earth (which would slip down if not supported) presses against the wall on the most unexceptional principles, CB acting perpendicularly against AE at K, m; D or 3 AE3. AB2 OBE2 where m denotes the specific gravity of the earth of the triangle ABE.-It may be remarked that this was deduced from using the area only of the profile, or transverse triangular section ABE, instead of the prisimatic solid of any given length, having that triangle for its base. And the same thing is done in determining the power of the wall to support the earth, viz, using only its profile or transverse section in the same plane or direction as the triangle ABE. This it is evident will produce the same result as the solids themselves, since, being both of the same given length, these have the same ratio as their transverse sections. In addition to this determination, we may here further observe, that this pressure ought to be diminished in proportion to the cohesion of the matter in sliding down the inclined plane BE. Now it has been found by experiments, that a body requires about one-third of its weight to move it along a plane surface. The above expression must therefore be reduced in the ratio of 3 to 2; by which means it becomes m for the true practical efficacious pressure of the earth against the wall. AE3. AB2 9BB2 AB BE Since which occurs in this expression of the force of the earth, is equal to the sine of the AEB to the radius I, put the sine of that Ee; also put a AE the altitude of the triangle; then the above expression of the force, viz, VOL. III. AE. AB2 S |