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ordinates by 1, Y1, 21; X2, Y2, 22; xs, Ys, 23; &c.; and the co-ordinates of their common centre of gravity by xe, Ye, za; then
Example.-Suppose that as the co-ordinate planes are taken, a horizontal plane in a ship's engine-room; a vertical longitudinal plane traversing that room amidships; and a transverse plane in a convenient assumed position; and that the weights under consideration are four portions of the engine, of the weights and in the positions stated in the following Table :
Then the following Table shows the calculation of the moments and resultant moments, and of the co-ordinates of the centre of gravity:
Divide by total weight, 2250, giving for Forward, 0.22 | Left, 0.33 | Up,3.11
the co-ordinates of the centre of gravity.
IV. To find the effect upon the position of the centre of gravity of a set of weights, of shifting one of those weights into a
new position, multiply the weight shifted by the distance through which it is shifted, and divide by the sum of all the weights: the quotient will be the distance through which the centre of gravity will be shifted, in a direction parallel to that in which the weight is shifted. "
V. To find how far a given single weight must be shifted, in order to shift the common centre of gravity through a given distance in the same direction, multiply the sum of the weights by the distance through which their common centre of gravity is to be shifted, and divide by the single weight.
For example, if weight No. 3 of the preceding table, 250 lbs., be shifted in any direction through 4 feet, the centre of gravity will be shifted in the same direction through
Centres of Gravity and Moments of Bodies.-The supposition of the weight of a body being concentrated at a point is a mathematical fiction, as has been already stated in Chap. III.; but the centre of gravity of a body, at which its weight may be conceived to be concentrated, without error as regards its mechanical action as a whole, can always be found.
When the body is homogeneous, or composed of material of uniform heaviness throughout, the following principles serve to determine the centre of gravity and the moment of its weight relatively to a given plane.
I. The centre of gravity of a homogeneous body is its centre of figure, or of mean distances.
II. The moment of the weight, or statical moment, of a homogeneous body, relatively to a given plane, is equal to the product of its geometrical moment relatively to that plane into the heaviness of the material.
The rules which have reference to plane areas are applicable to plates or prismatic bodies of uniform thickness, having those areas for bases. In computing statical moments by means of them, the following rule is to be used:
III. Multiply the geometrical moment of the plane base by the thickness, and by the heaviness of the material. The product of the thickness into the heaviness is the weight per unit of area. In all such calculations, if the dimensions are expressed in feet,
and the heaviness in lbs. to the cubic foot, the moment is expressed in foot-pounds. If the heaviness is that of sea-water, the geometrical moment itself is the statical moment in cubic feet of sea-water at a leverage of a foot; which may be multiplied by 64 for foot-pounds, or divided by 35 for foot-tons.
When a body is heterogeneous, or consists of parts of different heaviness, the following rule is to be applied :
IV. Divide the body into parts, each of which is of uniform or sensibly uniform heaviness; find the centre of gravity of each such part separately; conceive the weight of each part to be concentrated at its own centre of gravity, and treat those weights as detached weights (according to rules before given in this chapter). The Resultant of a Pressure distributed over a plane surface is found by the following rules :
I. If the intensity is uniform, muliply the area of the surface by the intensity.
II. If the intensity is not uniform, conceive that the surface lies horizontal, and that a solid stands upon it, whose height at each point represents the intensity of the pressure at that point. Then the volume of that solid will represent the amount of the pressure. As to finding the volume of such a solid, see Chapter IX.
The Centre of Pressure means, a point traversed by the resultant of a pressure that is distributed over a surface. When the surface pressed upon is plane, the centre of pressure is a point in that surface itself, and is found according to the following rules:
III. When the intensity of the pressure is uniform, the centre of figure is the centre of pressure.
IV. When the intensity of the pressure is not uniform, find the centre of the solid of Rule II., from which let fall a perpendicular on the pressed surface; the foot of that perpendicular will be the centre of pressure; or otherwise, find the co-ordi nates of the centre of that solid relatively to two axes in the plane of the pressed surface: these will be the co-ordinates of the centre of pressure.
V. If the pressure is that of a fluid upon a solid immersed on it, then, as already stated (Chaps. I., II., and III.), the Resultant Pressure is equal and opposite to the weight of the volume of fluid displaced; and the centre of pressure is the centre of that volume, or CENTRE OF DISPLACEMENT.
Displacement and Centre of Buoyancy-Curve and Scale of DisplacementMethods of Computing Displacement-Computation of Cross SectionsComputation of Water Sections--Computation of Displacement in LayersAppendages-Computation of Midship Section in Layers--Determination of Centre of Buoyancy.
DISPLACEMENT, AND CENTRE OF BUOYANCY.
THE drawings of a proposed ship, from which measurements for finding the displacement and stability of the ship are made, consists of three plans, viz.:-Sheer, Half-Breadth, and Body Plans. A great variety of lines are shown on these plans. Some of those lines are specially connected with the practical execution of the ship, and will be fully treated of in the following division of this work. In connection with displacement, two sorts of lines only have to be considered; and those are Water-lines and Vertical Cross Sections.
The water-lines of the immersed part of the ship are first drawn to the outside of plank, as represented in the accompanying plan of the U. S. steamer Antietam*, Plate 6, drawn on a scale of one-quarter of an inch to a foot, or 4th of the real dimensions, which scale is commonly employed. The 16th water-line is the load-water line. To obviate the necessity of having to take off the half-breadths of the ordinates in order to compute the displacement, centre of buoyancy and metacentre, a Table of the ordinates as taken off is appended.
A WATER-LINE is the outline of a horizontal section of the ship; being the line in which the surface of the water either actually meets the skin of the ship when she floats upright at a certain depth of immersion, or would meet the skin of the ship if she were to float to a certain supposed depth.
The vertical depth between the highest and lowest water-lines is divided into a number of equal intervals by the intermediate water-lines, which are numbered in succession 2 W. L., 4 W. L., 6 W. L., &c. If considered necessary in calculating the displacement, those intervals may be subdivided, according to the principles explained in Chapter VIII. This is often required at those parts
*The plan in the book is reduced from this scale.
of the vessel's bottom which are rapidly curved, such as the bilge, being the part which connects the nearly upright side with the comparatively flat floor. The plane horizontal area enclosed within a water-line is called a Water-section or a Plane of Flotation.
As a base-line, or longitudinal axis, for all the measurements of the ship, there is taken the Centre-Line of the Load-watersection, marked AB in the sheer and half-breadth plans. It is held to extend from the forward edge of the rabbet of the stem at A, to the after-edge of the rabbet of the stern-post at B, being the points where the surface of the vessel meets the stem and sternpost respectively; and that distance is divided into a sufficient number of equal intervals, which may be subdivided where the figure of the vessel requires it.
When the vessel has two stern-posts, a main stern-post and a rudder-post, as in the case of most screw steamers, it is at the foremost of the two, or main stern-post, that the length of the load-water-section is held to terminate. In the plan, the base-line is divided into twenty intervals from the centre of the length each way. In vessels having a full bow and stern, several of the foremost and aftermost intervals should be subdivided into half-inter vals, because of the rapid curvature of the water-lines. In vessels with fine entrances and runs to the water-lines, like the plan given, such subdivision is unnecessary. By calculating the displacement of the fore and after bodies separately, their relative capacities may be ascertained.
The ordinates of a ship are all half-breadths; that is to say, horizontal lines, measured from the central vertical longitudinal plane traversing the axis AB (Antietam's Plan) to the outside surface of the plank on the vessel. Each ordinate belongs at once to the water-section and to the vertical section of which it is the intersection; and by Simpson's rules, it has two multipliers, according as it is to be used in computing the area of the watersection, or that of the vertical section.
METHODS OF COMPUTING DISPLACEMENT.
There are two processes for computing the displacement of a ship, both of which should always be gone through; because the intermediate steps of both processes are necessary in the subsequent operation of finding the centre of buoyancy; and also because the agreements of their final results form a check on the accuracy of the calculations.