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Original volume of displacement = 100 × 20 × 10,

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and seeing that the weight of ship is the same, these two volumes must be the same, so that d = 12.5 ft.

It is important to note that if a compartment is filled with coals, stores, etc., the space thus occupied cannot be taken up by the water, and thus the lost buoyancy is much less. For instance, in the above vessel, if the central compartment were filled with coal from the ends, the vessel floating at the same draught of 10 ft. before bilging, the sinking after bilging would only be 9 in.

If the watertightness of either of the bulkheads in the former example ceased below a height of 12 ft. 6 in., the water would

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not be confined to the central compartment, and the vessel would sink. This illustrates the importance of carrying watertight bulkheads well above the waterline. This is always done with the principal bulkheads in vessels of the Royal Navy (see Figs. 52 and 54).

Watertight flats are important because they serve to confine the results of any damage which may occur. Thus, in the first of the above examples, if a watertight flat were worked 5 ft. from the keel between the bulkheads, if bilging took place

(i.) Below the flat, a sinkage of 1 ft. would result;
(ii.) Above the flat, a sinkage of 11 ft. would result.

The greater sinkage in the latter case is due to the fact that the waterplane area is reduced as well as the buoyancy.

CHAPTER XVII.

INITIAL STABILITY, METACENTRIC HEIGHT, ETC.

Centre of Gravity.-The weight of a body is the sum or resultant of the weights of all the particles composing it, and this resultant

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acts through a definite point however the body is placed. This point is termed the centre of gravity (C.G.), and the body being at rest, we can regard the whole weight as being concentrated at the centre of gravity.

If two weights each of 5 lbs. are placed, as in Fig. 160, 24 in. apart, the C.G. of the system must be midway between them at G, and if the weights were held by a string, it would be immaterial, so far as the

string was concerned, whether the weights are as shown, or the whole 10 lbs. concentrated at G. Again, if the weights are 10 and 5 lbs. respectively, 24 in. apart, we should need to support at the point G, 8 in. from the larger weight, i.e. at the centre of gravity.

The centre of gravity of a ship is the point at which we may regard the whole of the weight to be concentrated.

Centre of Buoyancy. The resultant of the upward buoyancy must have its line of action through the centre of gravity of the displacement. When the water filled the space, before the ship was there, the weight of the water acted through this point, and so the support of the surrounding water, or, as we term it, the buoyancy, must also act through the same point. This point is termed the centre of buoyancy, being the C.G. of the displacement.

The position of the centre of buoyancy (C.B.), relative to the

waterplane, can be accurately calculated by simple rules1 (which it is not proposed to discuss in this work), and this position has an important influence on the transverse stability of the ship. The following formula gives the approximate distance of the C.B. below the L.W.L., viz.

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where D = mean draught (excluding keel projection, if any)

W = displacement in tons

T= tons per inch

This formula is found to give results very close to the figures obtained from detailed calculation, as the following examples show:

1. Vessel 2135 tons, 187 tons per inch, 13 ft. 6 in. mean draught.

C.B. below L.W.L. approx. = 3 (13.5

+

2135

2 18.7 x 12

The actual calculation gave 5.37 ft.

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¿) = 5·42 ft.

2. Vessel 15,000 tons, 57-2 tons per inch, 26 ft. 9 in. mean draught.

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Conditions of Equilibrium of a Body floating freely and at Rest in Still Water.-We have seen that for any body floating freely and at rest, the upward support of the buoyancy exactly equals the weight of the body. The weight acts through the centre of gravity, and the buoyancy acts through the centre of buoyancy. If the vessel is in equilibrium, i.e. has no tendency of herself to move, these two forces which are acting on the ship must act in the same vertical line. That is to say, the C.G. and the C.B. must be in the same vertical line.

This is the condition of things in a tug of war. If no movement takes place, it is evident that both sides are equal, and they are both pulling in the same line.

We have thus two conditions which are satisfied in the case of a body floating in equilibrium, viz.—

1. The weight of the body exactly equals the weight of the water displaced, and

2. The C.G. and the C.B. must be in the same vertical line.

See the author's "Theoretical Naval Architecture."

Stable, Unstable, Unstable, and Neutral Equilibrium. - The equilibrium, however, may be either (a) stable, (b) unstable, or (c) neutral. These kinds of equilibrium are defined as follows, viz.

(a) Stable Equilibrium.—If the vessel be slightly inclined from her position of rest she will tend to return.

(b) Unstable Equilibrium.-If the vessel be slightly inclined from her position of rest, she tends to incline still further.

(c) Neutral Equilibrium.—If the vessel be slightly inclined from her position of rest, she neither tends to return to or to incline still farther from that position.

These three kinds of equilibrium are seen in Fig. 161, in which cylinders are shown resting on a smooth table.

(a) In the first case the C.G. is below the centre, and if the cylinder is

DOOOOO

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slightly inclined it is seen that the two forces acting, viz. (1) the weight through the C.G., and (2) the support of the table through the point of contact, form a couple which tend to take the cylinder back to the upright. This is a state of equilibrium which is stable.

(b) In the second case the C.G. is above the centre, and on slightly inclining it is seen that the couple acting is an upsetting couple, and the cylinder will incline still farther. This is a state of equilibrium which is unstable.

(c) In the last case the C.G. coincides with the centre of the figure, and there will be no tendency to return to or incline further from the original position on giving a small inclination. This is a state of equilibrium which is neutral. This is the necessary condition for a billiard ball. The ivory must be perfectly homogeneous, so that the C.G. is at the centre of the ball, and the ball must be perfectly spherical, so that the support will always act through the centre.

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In the case of a ship let Fig. 162 represent the vessel inclined to a small angle 0. WL was the position of the waterline on the ship when upright; B, the

position of the C.B. when upright; and G, the position of the C.G. The ship as inclined has a new waterline, W'L', and the C.B. of the new displacement will be at B', so that the buoyancy will now act through B'. Let the vertical through B' cut the original vertical through B in M.

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Then in the first case, in Fig. 161, we notice that the couple acting on the ship tends to bring her back to the upright. In the second case the couple tends to incline her still further from the upright. We see that whether the couple is a righting couple or an upsetting couple depends on the relative positions of the points G and M.

If G is below M, the couple is a righting couple, and the ship is in stable equilibrium. If G is above M, the couple is an upsetting couple, and the ship is in unstable equilibrium. If G coincides with M, the ship is in neutral equilibrium. The point M is thus seen to be an important point, as its position relative to G determines the state of the equilibrium in the upright condition. It is given the name of the transverse metacentre when dealing with transverse inclinations, and the distance between G and M is termed the metacentric height.

We thus see that there are three conditions which must be fulfilled in order that a floating body shall float freely and at rest in stable equilibrium, viz.

1. The weight of the body must exactly equal the weight of the water displaced;

2. The C.G. and the C.B. must be in the same vertical line; and 3. The C.G. must be below the transverse metacentre. For angles up to 10 or 15° the intersection of the vertical through B' with the original vertical through B is practically at the same point, viz. M. For larger angles this will not be the case.

Before dealing generally with the question of initial stability, i.e. stability at small angles, we shall deal separately with the two

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