is to the circumference where the power is applied: viz. as circumf. of D c to dist. B I. B 3. The endless screw, or perpetual screw, is one which works in, and turns a dented wheel D F, without a concave or female screw; being so called because it may be turned for ever, without coming to an end. From the diagram it is evident that while the screw turns once round, the wheel only advances the distance of one tooth. W H 4. If the power applied to the lever, or handle of an endless screw, A B, be to the weight, in a ratio compounded of the periphery of the axis of the wheel E н, to the periphery described by the power in turning the handle, and of the revolutions of the wheel D F to the revolutions of the screw с B, the power will balance the weight. Hence, 5. As the motion of the wheel is very slow, a small power may raise a very great weight by means of an endless screw. And therefore the chief use of such a screw is, either where a great weight is to be raised through a little space; or where only a slow gentle motion is wanted. For which reason it is very serviceable in clocks and watches. The screw is of admirable use in the mechanism of micrometers, and in the adjustments of astronomical and other instruments of a refined construction. 6. The mechanical advantage of a compound machine may be determined by analyzing its parts, finding the mechanical advantage of each part severally, and then blending or compounding all the ratios. Thus, if m to 1, n to 1, r to 1, and s to 1, show the separate advantages; then m nrs to 1, will measure the advantage of the system. d R 7. The marginal representation of a common construction of a crane to raise heavy loads, will serve to illustrate this. By human energy at the handle a, the pinion b is turned; that gives motion to the wheel w, round whose axle, c, a cord is coiled; that cord passes over the fixed pulley, d, and thence over the fixed triple block, B, and the moveable triple block, P, below which the load, L, hangs. Now, if the radius of the handle W P be 6 times that of the pinion, the radius of the wheel w 10 times that of its axle, and a power equivalent to 30 lbs. be exerted at a; then, since a triple moveable pulley gives a mechanical advantage of 6 to 1, we shall have 30 x 6 x 10 x 6=10800 lbs. and such would be the load, L, that might be raised by a power of 30 lbs. applied at a, were it not for the loss occasioned by friction.* SECTION IV.-General application of the principles of Statics to the equilibrium of Structures. Every structure is exposed to the operation of a system of forces; so that the examination of its stability involves the application of the general conditions of equilibrium. Now, no part of a structure can be dislocated, except it be either by a progressive, or a rotatory motion. For either this part is displaced, without changing its form, in which case it is as a system of invariable form, incapable of receiving any instantaneous motion, which is not either progressive or rotatory; or else it happens to be displaced, changing at the same time its form and this, considering the cohesion of tenacity, cannot take place, without the breaking of that part in its weakest section; which generates a progressive motion, if the force acts perpendicularly to the section; and a rotatory motion, if it acts obliquely. We shall here consider the most useful cases; indicating by the word stress, that force which tends to give motion to the structure; by resistance, that which tends to hinder it. Equilibrium of Piers. 1. Taking the marginal figure for the vertical section of a pier, we may reason upon that section instead of the pier itself, if it be of uniform structure. Z D G X P Let & be the place of the centre of gravity, SR Z the direction in which the stress acts, meeting x 1, the vertical line through the centre of gravity, in 1. Then, considering the stress as resolvable into two forces, one P, vertical, the other, q, horizontal; the pier (regarding it as one body) can only give way either by a We shall insert a selection of useful mechanical contrivances, after we have given the principles of dynamics. progressive motion from в towards A, or by a rotatory motion about A. 2. The progressive motion is resisted by friction. If w denote the weight of the pier, P the stress estimated vertically, and its horizontal effort, then the pressure on the base =w+P, and its friction = f (w + P), which is the amount of the resistance to progressive motion. So that to ensure stability in this respect we must have f(w +P) > Q... While, to ensure stability in regard to rotation, we must have W.AX P.AE > Q.ES (1) (2) 3. The second condition may be ascertained by a graphical process, thus: From the point A, let fall, on the direction of the stress, the perpendicular A z. Then, s being put for the whole stress, W. AX S. A Z. Or, suppose the two forces м and s to be applied at 1, and complete the parallelogram, having sides which represent these forces. Then must the diagonal produced meet the base on the side of A, towards в, to ensure stability. 4. If, as is very frequently the case, the vertical section of the pier is a rectangle, and s represent the specific gravity of the material of which it is constituted; then the condition of the two kinds of equilibrium will be denoted by these two equations: viz. f. CB. AB. S = Q (3) . . . . A B2 s = 2 q.. (4) • ... Example. Suppose a rectangular wall 39-4 feet high, and of a material whose specific gravity is 2000, is to sustain a horizontal strain of 9900 lbs. avoirdupois at its summit on the unit of length, 1 foot: what must be the thickness that there may be an equilibrium, taking ƒ = . Here, that the wall may not be displaced horizontally, we must have And 2dly, that it may not be overturned, we must have Here, as the thickness required to prevent overturning is much the greatest, the computation in reference to the other kind of equilibrium may usually be avoided. nated the line of rupture or the natural slope, or natural declivity. In sandy or loose earth, the angle B A G seldom exceeds 30°; in stronger earth it becomes 37° and in some favourable cases more than 45°. 2. Now, the prism whose vertical section is DAG, has a tendency to descend along the inclined plane & A by reason of the force of gravity, g; but it is retained in its place, 1st, by the force, q, opposed to it by the wall, and 2dly, by its cohesive attachment to the face A G, and by its friction upon the same surface. Each of those forces may be resolved into one, which is perpendicular to G A, and which is inoperative as to this inquiry, and into another whose action is parallel to GA. The lines PI and I н, represent these composants of p, that force being represented by the vertical line P H, drawn from the centre of gravity P of the prism. The direction of the force q is represented by the horizontal line Q H, and its composants by the lines Q L, H L. The force that gives the triangle its tendency to descend is I H; and the force opposed to this is L H together with the effects of cohesion and friction. Thus, IH = LH + cohesion + friction. It is evident, therefore, that the solution to this inquiry must be, in great measure, experimental. 3. It has been found, however, theoretically, by M. Prony,* and confirmed experimentally, that the angle formed with the vertical by the prism of earth that exerts the greatest horizontal stress against a wall, is half the angle which the natural slope of the earth makes with the vertical: and this curious result greatly simplifies the whole inquiry. The state of equilibrium is expressed by this equation: viz. AD. AE.s= A D3. S. tan' DA G. s and s representing the specific gravities of the wall and earth respectively. Example. The wall to be 39.37 feet high, of brick, specific See a demonstration at p. 369, vol. ii. tenth edition of Dr. Hutton's Course of Mathematics. gravity 2000, and the terrace of strong earth specific gravity 1428, natural slope. 53° from vertex. Then the above equation becomes x2 × 2000 × 39.37 x 39.373 x 1428 x tano 26 = or x 39.37 tan 26° 1428 39.37 11428 6000 =19-685 x 4878-9.6 feet, thickness of the wall. 4. Of the experimental results the best which we have seen are those of M. Mayniel, from which the following are selected; all along supposing the upper surface of the earth and of the wall which supports it, to be both in one horizontal plane. 1st. Both theory and experiment indicate that the resultant HQ of the thrust of a bank, behind a vertical wall, is at a distance A Q from the bottom of the wall AD, the height. 2dly. That the friction is half the pressure, in vegetable earths, four-tenths in sand. 3dly. The cohesion which vegetable earths acquire, when cut in turfs, and well laid, course by course, diminishes their thrust by full two-thirds. 4thly. The line of rupture behind a wall which supports a bank of vegetable earth is found at a distance D G from the interior face of the wall equal to 618 h, h being the height of the wall. G='677 h. 5thly. When the bank is of sand, then D G= 6thly. When the bank is of vegetable earth mixed with small gravel, then D G='646 h. 7thly. If it be of rubbles, then D G='414 h. Sthly. If it be of vegetable earth mixed with large gravel, then D G 618 h. Thickness of Walls, both faces vertical. 1. Wall brick, weight of cubic foot = 109 lbs. avoird, bank vegetable earth, carefully laid, course by course, D F='16 h. 2. Wall unhewn stones, 135lbs. per cubic foot, earth as before, D F='15 h. 3. Wall brick, earth clay well rammed, D F•17 h. 4. Wall unhewn stones, earth as above, D F 16 h. 5. Wall of hewn free stone, 170 lbs. to the cubic foot, bank vegetable earth, D F 13 h; if the bank be clay D F='14 h. 6. Bank of earth mixed with large gravel, Wall of bricks . . .. unhewn stone hewn free stone 19 h. D F=17 h. D F='16 h. |