148

THE CENTRE OF BUOYANCY AND

94. The Conditions of Stability of a Ship.

In addition to satisfying the conditions of equilibrium of a floating body (§§ 45, 48, 84) it is necessary that a ship should fulfil the further condition of stability of equilibrium, so as not to capsize; if slightly disturbed from the position of equilibrium, the forces called into action must be such as tend to restore the ship to its original position.

Practically the stability of a ship is investigated by inclining it; weights are moved across the deck and the angle of heel thereby produced is observed, and thence an estimate of the stability can be formed.

95. Let the total weight of the ship be W tons, and let it displace V ft3 of water; then

W=wV, or 2240 W=DV,

where w denotes the density of the water in tons/ft3, and Dn lb/ft3, so that D=2240w.

Now suppose a weight of P tons on board, originally amidships, is moved to one side of the deck, a distance of bft; and that the ship, originally upright, is observed to heel through an angle 0 (fig. 38).

THE METACENTRE OF A SHIP.

149

The C.G. of the displaced water, called the centre of buoyancy (C.B.), will move on a curve (or surface) called the curve (or surface) of buoyancy, from B to B2, such that GB, is vertical in the new position of equilibrium, G2 being the new C.G. of the ship when P tons is moved from 9 to 92, so that the ship will move as if the surface of buoyancy was supported by a horizontal plane.

As P is moved across the deck from g to g2 a distance of b ft, so the C.G. of the body moves on a parallel line from G to G2, such that GG2=bP/W; this follows because the moments of P and W about Gg must be the same; and, if the new vertical B2G2 cuts the old vertical BG in m,

P

2

Gm=b cot 0, G2m=
bcot

W

The ultimate position M of m for a small angle of heel is the point of ultimate intersection of the normals at B and at the consecutive point B on the curve of buoyancy, and M is therefore the centre of curvature of the curve of buoyancy at B; the point M is called the metacentre, and GM is called the metacentric height.

In the diagram the ship is drawn for clearness in one position, and the water line is displaced; but the page can be turned so as to make the new water line horizontal.

92

96. For stability of equilibrium the metacentre M must be above G, for if M were below G then on bringing P back suddenly from g2 to g, the forces acting on the ship would form a couple tending to capsize the ship; but if M is above G the forces would then form a couple, consisting of W acting vertically downwards through G, and W acting vertically upwards along B2m, tending to restore the ship to the upright position.

150

METACENTRIC HEIGHT.

The angle of heel is measured either by a spirit level or by the deflection of a pendulum or plummet.

If the ship is symmetrical and upright when P is amidships, and if moving the weight P tons across the deck through 26 ft causes the plummet to move through 2a ft when suspended by a thread 7 ft long, then

and G2m may be taken as the metacentric height GM. Thus in H.M.S. Achilles, of 9000 tons displacement, it was found that moving 20 tons across the deck, a distance of 42 ft, caused the bob of a pendulum 20 ft long to move through 10 inches.

5

Here W=9000, P=20, b=21, l=20, a=12; and therefore GM=2.24 ft. Also sin 0=0·02083, 0=1° 12'.

97. The displacement W tons or V ft3 is determined by approximate calculations from the drawings of the ship, as also the C.B. B; while G is determined from the weights of the different parts of the structure, and from the distribution of the cargo and ballast.

If the weight P was hoisted vertically up the mast a distance h, B and M would not change, but G would ascend to G1, through a height GG1=hP/W ft.

The metacentric height would be correspondingly diminished; so that if P is now moved along a yard on the mast a distance b feet, the ship will heel through an angle 1, such that

Pb= W. GM sin 0 W. GM sin 01;

=

and to produce the original angle of heel 0, P requires to be moved through a less distance b', such that

b-bh sin 0.

ALTERATION OF TRIM.

151

98. So far we have considered as the chief practical problem the transverse metacentre M in its relation to the heeling or rolling of a ship; but a similar metacentre exists for any alteration of trim, caused either by change of stowage of cargo, or by press of sail and other propulsion; this longitudinal metacentre is found by a similar experimental process, but from the shape of a ship it is necessarily much higher than the transverse

metacentre.

The trim of a ship is defined as the difference of draft of water at the bow and stern, and the change of trim is defined as the sum of the increase of draft at one end and the decrease of draft at the other.

Suppose the trim is changed x inches in a vessel L ft long at the water line by moving P tons longitudinally fore and aft through a ft, and that the ship turns through a small angle 0, a gradient of one in 12L/x.

The moment to change the trim is Pa ft-tons, so that if M, denotes the longitudinal metacentre,

Pa= W.GM,.sin 0 W. GM,.x/12L;

thus the moment required to change the trim one inch is W. GM/12L ft-tons.

For instance, if a ship of 9200 tons, 375 ft long, has a longitudinal metacentric height of 400 ft, and a weight of 50 tons already on board is shifted longitudinally through 90 ft, the change of trim will be about 5 inches.

Practically it is found convenient to incline the ship by filling alternately the boats suspended on either side of the ship with a known weight of water; and to change the trim by filling and emptying water tanks at the ends of the ship.

152 WEDGES OF EMERSION AND IMMERSION.

99. As the ship heels through an angle 0, and the water line changes from LL' to L2L1⁄2 (fig. 38), a certain volume U of water may be supposed removed from the wedgeshaped volume L'FL1⁄2, called the wedge of emersion, to the volume LFL2, called the wedge of immersion, so as to form the new volume of displacement L2KL1⁄2.

If b1, b2 denote the c.G.'s of the wedges of emersion and immersion, BB, is parallel to b1b2; and if BY, c12 are the projections of BB, bb, on the new water line LL2,

We notice that when is evanescent, the line b1b2 coalesces with the water line LL; and therefore the tangent to the curve of buoyancy at B, being the ultimate direction of the line BB2, is parallel to the water line LL', a theorem due to Bouguer.

If the C.G. of the ship is at G, and GZ is drawn perpendicular to B2m, the moment in ft-tons of the couple tending to restore the ship to the upright position is W.GZ=W(BY— BG sin 0)

Atwood's formula (Phil. Trans., 1798).

Curves are now drawn for ships by naval architects, called cross curves of stability, which exhibit graphically the value of the righting moment W. GZ for a given inclination , say an angle of 30°, 45°, or 60°, and for different drafts of water of the ship and displacements W or V; the volumes U of the wedges of immersion and emersion are calculated and the corresponding values of C12; also the values of BG, and thence GZ is known for an assumed position of G.