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what distance the point G is to be placed, we must multiply twice the radius CE by the chord AB, and divide the product by thrice the length of the arc AEB. The quotient

E

is the distance CG from the centre C of the circle of the centre of gravity of the sector. 1279. To find the centre of gravity of the crown portion of an arch DAEBF (fig. 540.) comprised between two concentric axes, we must

1. Find the centre of gravity A of the greater sector AEBC, and that of the smaller one DFG.

2. Multiply the area of cach

of these sectors by the distance

E

C

Fig. 539.

of their respective centres of gravity from the common centre C.

C

Fig. 510.

B

F

3. Subtract the smaller product from the greater, and divide the remainder by the area of DAEBF; the quotient will give the distance of the centre of gravity G from the centre C.

E

1280. To determine the centre of gravity of the segment AEB; subtract the product of the area of the triangle ABC (fig. 541.) multiplied by the distance of its centre of gravity from the centre C, from the product of the area of the sector, by the distance of its centre of gravity from the same point C, and divide the remainder by the area AEB; the quotient expresses the distance of the centre of gravity G of the segment from the centre C, which is to be set out on the radius, and A which divides the segment into two equal parts.

Fig. 541.

B

It would, from want of space, be inconvenient to give the strict demonstrations of the above rules; nor, indeed, is it absolutely necessary for the architectural student. Those who wish to pursue the subject au fond, will, of course, consult more abstruse works on the matter. We will merely observe, that whatever the figure whose centre of gravity is sought, it is only necessary to divide it into triangles, sectors, or segments, and proceed as above described for the pentagon, fig. 528.

OF THE CENTRE OF GRAVITY OF SOLIDS.

1281. It is supposed in the following considerations, that solids are composed of homogeneous particles whose weight in every part is uniform. They are here arranged under two heads, regular and irregular.

1282. Regular solids are considered as composed of elements of the same figure as their base, placed one upon the other, so that all their centres of gravity are in a vertical line, which we shall call the right axis. Thus parallelopipeds, prisms, cylinders, pyramids, cones, conoids, spheres, and spheroids have a right axis, whereon their centre of gravity is found.

1283. In parallelopipeds, prisms, cylinders, spheres, spheroids, the centre of gravity is in the middle of the right axis, because of the similarity and symmetry of their parts equally distant from that point.

1284. In pyramids and cones (figs. 542, 543.), which diminish gradually from the base to the apex, the centre of gravity is at the distance of one fourth of the axis from the base.

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S

Fig. 542.

S

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Fig. 513

the quotient will be the distance of the centre of gravity G of the part of the truncated

cone or pyramid from its apex.

1286. The centre of gravity of a hemisphere is at the distance of three eighths of the radius from the centre.

1287. The centre of gravity of the segment of a sphere (fig. 544.) is found by the following proportion as thrice the radius less the thickness of the segment is to the diameter less three quarters the B thickness of the segment, so is that thickness to a fourth term which expresses the distance from the vertex to the centre of gravity, set off on the radius which serves as the axis.

1288. Thus, making r= the radius, e the thickness of the segment, and x = the distance sought, we have, according to La Caille,

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Fig. 14.

Suppose the radius to be 7 feet, the thickness of the segment 3 feet, we shall have

8x7x3-3x9

x= 12x7-3x4' which gives x=1+3=1+}}, equal the distance

of the centre of gravity from its vertex on the radius.

D

A

E

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D

1289. To find the centre of gravity of the zone of a sphere (fig. 545.), the same sort of operation is gone through as for truncated cones and pyramids; that is, after having found the centre of gravity of the segment cut off, and that in which the zone is comprised, multiply the cube of each by the distance of its centre of gravity from the apex A, and subtract- B ing the smaller from the larger product, divide the remainder by the cube of the zone. Thus, supposing, as before, the radius AC=7, the thickness of the zone 2, and that of the segment cut off = 14, we shall find the distance from the vertex of the centre of gravity of this last by the formula =), which in this case gives x= +x2x7x1-3x2; and pursuing the investigation, we have x=1 4(21--13) which will be the distance of the centre of gravity from the vertex A. of gravity of the segment in which the zone is comprised will, according to the same formula, ber= 8x7x33x12, which gives x=2+ ¦ for the distance of the centre of gravity

43x7-3.

from the same point A.

Sre-See

103

Fig. 515.

That of the centre

1290. The methods of finding the solidities of the bodies involved in the above inves tigation are to be found in the preceding section, on Mensuration.

OF THE CENTRE OF GRAVITY OF IRREGULAR SOLIDS.

A

C

B

D

E

K

1291. As all species of solids, whatever their form, are susceptible of division intc pyramids, as we have seen in the preceding observations, it follows that their centres of gravity may be found by following out the instructions already given. Instead of two lines at right angles to each other, let us suppose two vertical planes NAC, CEF (fig. 546.), between which the solid G is placed. Carrying to each of those planes the momenta of their pyramids, that is, the products of their solidity, and the distances of their centres of gravity, divide the sum of these products for each plane by the whole solidity of the body, the quotient will express the distance of two other planes BKL, DHM, parallel to those first named. Their intersection will give a line IP, or an axis of equilibrium, upon which the centre of gravity of the solid will be found. To determine the point G, imagine a third plane NOF perpendicular to the preceding ones, that is, horizontal; upon which let the solid be supposed to stand. of this plane let the momenta of the pyramids be found by also multiplying their solidity by the distance of their centres of gravity. Lastly, dividing the sum of these products by the solidity of the entire body, the quotient gives on the axis the distance PG of this third plane from the centre of gravity of the irregular solid.

Mechanically, where two of the surfaces of a body are parallel, the mode of finding the centre of gravity is simple. Thus, if the body be hung up by any point A (figs. 547, 548.), and a plumb line AB be suspended from the same point, it will pass through

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the centre of gravity, because that centre is not in the lowest point till it fall in the plumb line. Mark the line AB upon it; then hang the body up by any other point D, with a plumb line DE, which will also pass through the centre of gravity, for the same reason as before. Therefore the centre of gravity will be at C, where the lines cross each other. 1292. We have, perhaps, pursued this subject a little further than its practical utility in architecture renders necessary; but cases may occur in which the student will find our extended observations of service.

OF THE INCLINED PLANE.

1293. That a solid may remain in a perfect state of rest, the plane on which it stands must be perpendicular to the direction of its gravity; that is, level or horizontal, and the vertical let fall from its centre of gravity must not fall out of its base.

1294. When the plane is not horizontal, solids placed on it tend to slide down or to

overturn.

1295. As the surfaces of bodies are more or less rough, when the direction of the centre of gravity does not fall without their base, they slide down a plane in proportion to their roughness and the plane's inclination.

1296. Thus a cube of hard freestone, whose surfaces are nicely wrought, does not slide down a plane whose inclination is less than thirty degrees; and with polished marbles the inclination is not more than fifteen degrees.

1297. When a solid is placed on an inclined plane, if the direction of the centre of gravity falls without its base, it overturns if its surfaces are right surfaces, and if its surface is convex it rolls down the plane.

1298. A body with plane surfaces may remain at rest after having once overturned if the surface upon which it falls is sufficiently extended to prevent its centre of gravity falling within the base, and the inclination be not so great as to allow of its sliding on.

1299. Solids whose surfaces are curved can only stand upon a perfectly horizontal plane, because one of the species, as the sphere, rests only on a point, and the other, as cylinders and cones, upon a line; so that for their continuing at rest, it is necessary that the vertical let fall from their centre of gravity should pass through the point of contact with and be perpendicular to the plane. Hence, the moment the plane ceases to be horizontal the direction of the centre of gravity falls out of the point or line of contact which serves as the base of the solid, and the body will begin to roll; and when the plane on which they thus roll is of any extent they roll with an accelerated velocity, equal to that which they would acquire in falling directly from the vertical height of the inclined plane from the point whence they first began to roll.

1300. To find the force which is necessary to support a convex body upon an inclined plane, we must consider the point of contact F (figs. 549, 550.) as the fulcrum of an an

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gular lever, whose arms are expressed by the perpendiculars drawn from the fulcrum to the direction of the force CP and the weight CD, which in the case of fig. 549., where the force which draws the body is parallel to the plane,

P: N:: FC: FD.

Now as the rectangular triangle CFD is always similar to the triangle OSH, which forms the plane inclined by the vertical SO and the horizontal line OH, the proportion will stand as follows:

P: N::OS: SH.

In the first case, to obtain an equilibrium, the force must be to the weight of the body as the height OS of the inclined plane to its length SH.

1301. In the case where the force is horizontal (fig. 550.) we have, similarly,

P: N:: FA: FD,

and P: N::OS: OH.

In this last case, then, the force must be to the weight of the solid in proportion to the height

OS of the inclined plane to its base OH. In the first case the pressure of the solid on the plane is expressed by OH, and in the second by SH: hence we have ——

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In the first case it must be observed, that the effect of the force being parallel to the inclined plane, it neither increases nor diminishes the pressure upon that plane; and this is the most favourable case for keeping a body in equilibrio on an inclined plane. In the second case, the direction forming an acute angle with the plane uselessly augments the load or weight. Whilst the direction of the force forms an obtuse angle with the inclination of the plane, by sustaining a portion of the weight, it diminishes the load on the plane, but requires a greater force.

1302. The force necessary to sustain upon an inclined plane a body whose base is formed by a plane surface depends, as we have already observed, on the roughness of the surfaces, as well of the inclined plane as of the base of the body; and it is only to be discovered by experiment.

1303. Of all the means that have been employed to estimate the value of the resistance, known under the name of friction, the simplest, and that which seems to give the truest results, is to consider the inclination of the plane upon which a body, the direction of whose centre of gravity does not fall out of the base, remains in equilibrio, as a horizontal plane; after which the degrees of inclination may begin to be reckoned, by which we find that a body which does not begin to slide till the plane's inclination exceeds 30 degrees, being placed on an inclined plane of 45, will not require a greater force to sustain it than a convex body of the same weight on an inclined plane of 15 degrees.

1304. All that has been said on the force necessary to retain a body upon an inclined plane, is applicable to solids supported by two planes, considering that the second acts as a force to counterpoise the first, in a direction perpendicular to the second plane.

1305. When the directions of three forces, PG, QG, GR, meet in the same point G (fig. 551.), it follows, from the preceding observations on the parallelogram of forces, that to be in equilibrium their proportion will be expressed by the three sides of a triangle formed by perpendiculars to their directions; whence it follows, that if through the centre of gravity G of a solid, supported by two planes or by some other point of its vertical direction, lines be drawn perpendicular to the directions of the forces, if equilibrium exist, so will the following proportion, viz. P: Q: R::BA: BC : AC.

A

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Fig. 551.

1306. Lastly, considering that in all sorts of triangles the sides will between each other be as the sines of their opposite angles, we shall have P: Q: R::sin. BCA sin. BAC: sin. ABC; and as the angle BCA is equal to the angle CAD, and CBA to BAE, we shall have P: Q: R:: sin. CAD: sin. BAC sin. BAE; that is, that the weight is represented by the sine of the angle formed by the two inclined planes, and that the pressures upon each of these planes are reciprocally proportional to the sines of the angles which they form with the horizon.

THE WHEEL AND AXLE.

B

D

1307. The wheel and axle, sometimes called the axis in peritrochio, is a machine consisting of a cylinder C and a wheel B (fig. 552.) having the same axis, at the two extremities of which are pivots on which the wheel turns. The power is applied at the circumference of the wheel, generally in the direction of a tangent by means of a cord wrapped about the cylinder in order to overcome the resistance or elevate the weight. Here the cord by which the power P acts is applied at the circumference of the wheel, while that of the weight W is applied round the axle or another small wheel attached to the larger, and having the same axis or centre C. Thus BA is a lever moveable about the point C, the power P always acting at the distance BC, and the weight W at the distance CA. Therefore P: W::CA: CB. That is, the weight and power will be in equilibrio when the power P is to the weight W reciprocally as the radii of the circles where they act, or as the radius of the axle CA, where the weight hangs, to the radius of the wheel CB, where the power acts; or, as before, P: W:: CA: CB.

P

Fig. 552.

1308. If the wheel be put in motion, the spaces moved through being as the circum

ferences, or as the radii, the velocity of W will be to the velocity of P as CA to CB; that is, the weight is moved as much slower as it is heavier than the power. Hence, what is gained in power is lost in time; a property common to machines and engines of every class. 1309. If the power do not act at right angles to the radius CB, but obliquely, draw CD perpendicular to the direction of the power, then, from the nature of the lever, P: W::CA: CD.

BK

D

1310. It is to the mechanical power of the wheel and axle that belong all turning or wheel machines of different radii; thus, in the roller turning on the axis or spindle CE (fig. 553.) by the handle CBD, the power applied at B is to the weight W on the roller, as the radius of the roller is to the radius CB of the handle. The same rule applies to all cranes, capstans, windlasses, &c.; the power always being to the weight as is the radius or lever at which the weight acts to that at which the power acts; so that they are always in the reciprocal ratio of their velocities. To the same principle are referable the gimlet and auger for boring holes.

E

W

Fig. 553.

C

1811. The above observations imply that the cords sustaining the weights are of no sensible thickness. If they are of considerable thickness, or if there be several folds of them over one another on the roller or barrel, we must measure to the middle of the outermost rope for the radius of the roller, or to the radius of the roller must be added half the thickness of the cord where there is but one fold.

1312 The power of the wheel and axle possesses considerable advantages in point of convenience over the simple lever. A weight can be raised but a little way by a simple

lever, whereas by the continued turning of the wheel and axle a weight may be raised to any height and from any depth.

1313. By increasing the number of wheels, moreover, the power may be increased to any cxtent, making the less always turn greater wheels, by means of what is called tooth and pinion work, wherein the teeth of one circumference work in the rounds or pinions of another to turn the wheel. In case, here, of an equilibrium, the power is to the weight as the continual product of the radii of all the axles to that of all the wheels. So if the power P (fig. 554.) turn the wheel Q, and this turn the small wheel or axle R, and this turn the wheel S, and this turn the axle T, and this turn the wheel V, and this turn the axle X, which raises the weight W; then P: W::CB. DE. FG: AC. BD. EF. And in

[graphic]

Fig. 554.

the same proportion is the velocity of W slower than that of P.

Thus, if each wheel

be to its axle as 10 to 1, then P: W::13 103, or as 1 to 1000. Hence a power of one pound will balance a weight of 1000 pounds; but when put in motion, the power will move 1000 times faster than the weight.

1314. We do not think it necessary to give examples of the different machines for raising weights used in the construction of buildings: they are not many, and will be hereafter named and described.

OF THE PULLEY.

1315. A pulley is a small wheel, usually made of wood or brass, turning about a metal axis, and enclosed in a frame, or case, called its block, which admits of a rope to pass freely over the circumference of the pulley, wherein there is usually a concave groove to prevent the rope slipping out of its place. The pulley is said to be fixed or moveable as its block is fixed or rises and falls with the weight. An assemblage of several pulleys is called a system of pulleys, of which some are in a fixed block and the rest in a moveable one.

1316. If a power sustain a weight by means of a fixed pulley. the power and weight are

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