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AND ITS STABILITY.

353

The investigation of the stability is similar to that required for a ship aground (§ 142); the bell is supposed slightly displaced through an angle from the vertical position, represented in fig. 74, p. 355, by drawing the surface of the water and the vertical forces in their displaced position relative to the bell.

Now, if K denotes the point of attachment of the chain, G the C.G. of the metal of the bell, supposed homogeneous, B and B2 the c.G.'s of the volume occupied by the air in the upright and inclined positions, and M the metacentric centre of curvature of the curve of buoyancy BB2; then, as in § 101,

BM= Ak2/V,

where Ak denotes the moment of inertia, in ft, of the water area A about its c.G. F, and V denotes the volume, in ft3, of the air in the bell; but now the metacentre M lies below the centre of buoyancy B.

The forces acting upon the bell in the displaced position (i.) W

W

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lb, acting downwards through G;

(ii.) DV lb, acting upwards through M;

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-DV lb, the tension of the chain, acting

upwards through K;

form a couple, whose moment round K, dropping the factor sin 0, is in ft-lb,

W(1-1)KG-DV.KM

= W(1 − 1)KG_DV. KB-DA%2 ;

and the equilibrium of the bell is stable if this moment

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354

THE DIVING BELL

Suppose the bell is initially full of water, and that a small volume of air is pumped in; the bell will not continue to hang vertically unless

W(1–1)KG-DAk2 is positive.

But in working the bell the water is expelled, and the air escapes under the lower edge; although the air would force its way out through a hole at the top of the bell, but then the level of water in the bell could not be kept steady.

The stability of the bell, as measured by the above. righting moment, diminishes as air is pumped in and V increases, Ak2 remaining constant if the interior of the bell is cylindrical; so that stability must be secured when the bell is full of air.

At a great depth, where the density p of the air in the bell becomes appreciable, D in the formula must be replaced by D-p.

257. The original idea of the Diving Bell is of great antiquity (Berthelot, Annales de Chimie et de Physique, XXIV., 1891); but Smeaton was the first to use it for Civil Engineering operations in 1779; and it was extensively employed on the wreck of the Royal George, in 1782 and 1817; and in Ramsgate Harbour by Rennie in 1788.

But the Diving Bell is now generally superseded by the Diving Dress (fig. 75), which is an india-rubber suit for the diver, provided with a copper helmet fitted with small circular windows and an air valve.

The diver has thick lead soles on his boots, and carries leaden weights round his neck, so adjusted that his apparent weight in water and stability are nearly the same as ordinarily on land.

AND DIVING DRESS.

355

Fresh air is pumped down to him through a tube as in a diving bell, the air escaping through the valve at the back of his head; if the diver wishes to rise he partly closes the valve, which causes the dress to be inflated and increases his buoyancy; so also sunken vessels are raised nowadays by large india-rubber bags placed in the hold and pumped full of air.

In the Fleuss system the diver carries with him a vessel of compressed air, and he is thus independent of the pipe and can travel long distances; as was required, for instance, during the construction of the Severn tunnel, when the water burst in and flooded the workings.

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A form of diving bell is shown in fig. 76 which is useful for the construction of harbours; the neck of the bell reaches the surface of the water, and entrance is made to the bell through an air lock; a large diving bell of this nature was employed at the construction of the New Port of La Rochelle (Cosmos, 26th April, 1890).

356

SUBAQUEOUS OPERATIONS.

The upward thrust of the air being equal to the weight of a cylindrical column of the water on an equal base, the weight of this bell must exceed the weight of water it displaces.

The same principles are employed in sinking caissons for underwater foundations, as in the Forth Bridge (fig. 77); or in driving a tunnel under a river through muddy soft soil, as in the Hudson River and the Blackwall tunnels, now in progress. Air is forced in to equalize the pressure of the head of water, and to prevent its entrance, being retained by air locks through which the workmen and materials can pass; a slight diminution of air pressure allows the water to percolate sufficiently to loosen the ground, but an increase of pressure is apt to allow the air to blow out in a large bubble.

This system, due to Mr. Greathead, has overcome the difficulties of subaqueous tunnelling; but if employed in the projected Channel Tunnel, a pressure of about 10 atmospheres would be required, to which the workmen are not yet accustomed.

(The Diving Bell and Dress, J. W. Heinke, Proc. Inst. Civil Eng. XV.;

Diving Apparatus, W. A. Gorman, Proc. Inst. Mechanical Engineers, 1882;

The Forth Bridge, Engineering, Feb., 1890.)

Examples.

(1) Two thin cylindrical gasholders which will hold four times their weight of water, and one of which just fits over the other, will float mouth downwards half immersed in water.

If the larger one is now placed over the smaller, determine the position of equilibrium.

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(2) Two equal cylindrical gasholders, of weight W and height a, float with a length ma occupied by gas, which at atmospheric pressure would occupy a length a. If a weight P is placed upon one of them, and gas is transferred to the other till the top of the first just reaches the water, prove that the other rises a height

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(3) Prove that gas of constant pressure, measured by a height h of water, can be delivered by a gasholder in the form of a truncated cone, whose sides are inclined at an angle a to the vertical, if the thickness of the sides is h sin ɑ.

(4) Coal gas, of density 0-6 of that of the air, is delivered to the pipes at a pressure of 2 ins of water; prove that 300 ft higher the pressure will be given by 3.8 ins; the temperature being 10°C.

(5) Prove that the small vertical oscillations of a cylindrical solid, closed at the top and inverted over mercury in a wide basin, will synchronize with a pendulum of length

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where Mg denotes the weight of the body, a and B cm2 the horizontal cross sections of the interior and of the material of the body, σ and h the density of mercury and the height of the barometer, V cm3 the volume of air in the cavity, and z cm the difference of level of the mercury inside and outside.

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