This acts prejudicially in regard to the stability, as the following example will show :

Two vessels have been taken, Fig. 191, both with 13 ft. draught, 9 ft. freeboard, 32 ft. beam, and 2 ft. metacentric height. In one case, however, the side falls in as shown to the deck, where the breadth is 24 ft. This results in a much lower curve of stability than is obtained without the fall in; we have seen that the area enclosed by a curve of stability is of great importance.

The corresponding metacentric diagrams are also interesting. In the second case, directly the side falls in the metacentric curve drops as the draught increases. Such a ship if damaged so that a bodily sinkage results might very possibly lose her initial stability, owing to this drop of the metacentre as the draught increases.

H.M.S.

STATEMENT OF METACENTRIC HEIGHTS AND STABILITY, BASED ON AN INCLINING EXPERIMENT MADE ON THE SHIP ON MARCH 27, 1900.

A. The ship when fully equipped, with reserve
feed-tanks empty, with 300 tons of coal
in lower bunkers and 300 tons of coal in
upper bunkers, at a mean draught of 20
ft. 6 in. has a metacentric height of
B. The ship when fully equipped, with fresh-
water and reserve feed-tanks and all
bunkers quite full, i.e. with 1085 tons
of coal on board, at a mean draught of 21
ft. 10 in. has a metacentric height of...
When lightened to a mean draught of 18 ft. 3
in., or when boilers are full to working
height, engine condensers and feed-tanks
at working height, and all coals, water
(including reserve feed), provisions,
officers' stores, and one half warrant
officers' and engineers' stores consumed,
the metacentric height is about

2.4

2.4

2:0

Remarks.

NOTE. So far as stability is concerned the coal may be worked in any manner desired by the Commanding Officer.

CHAPTER XX.

THE ROLLING OF SHIPS.

Rolling in Still Water.-Rolling in still water is of no immediate practical importance, because under ordinary circumstances a ship will not roll in still water. It is, however, necessary to study the subject, because it is only when the conditions operating in this case are understood that we are able to extend the inquiry to the more difficult case of rolling among waves.

If a ship, floating upright in stable equilibrium in still water, is inclined to a certain angle from the upright, the couple tending to take her back to the upright is W × GZ. If the ship is released she will acquire angular velocity, passing through the upright to an angle on the other side rather less than 0. At this new angle, ', say, she will have a couple tending to take her back to the upright, and so the ship, once being inclined and released, will continue to oscillate through smaller and smaller arcs of oscillation until she finally comes to a position of rest. When the ship is inclined to the angle 0 as above, it is necessary to do work on the ship to effect the inclination, and this work is stored up in the ship as potential energy, or energy due to position. When the ship is released, this energy becomes converted into other forms of energy, and, if no resistances were operating tending to stop the motion, the ship, when passing through the upright, would have kinetic energy, or energy due to motion, exactly equal to the original potential energy.

This conversion of energy of one form into energy of other forms is well seen in the case of a stone on the top of a house. In some way work has had to be done to get the stone there, and the stone, in virtue of its position, has stored up in it a certain amount of potential energy. If the stone is released, it will reach the ground with a certain velocity, and (neglecting the friction of the air) the kinetic energy then possessed is equal to the original potential energy. When stopped by the ground both potential energy and kinetic energy disappear, but the energy is not lost, but is dissipated into the form of heat energy.

In the case of a ship rolling unresistedly the energy is alternately potential and kinetic at the extremity and middle of each roll, and the rolling would go on continuously; but when resistances operate, the energy gets drained away from the ship and becomes finally dissipated by imparting heat and motion to the air and water surrounding the ship.

Unresisted Rolling in Still Water.-It can be shown that for unresisted rolling in still water the period of a single oscillation (from port to starboard, or vice versa) is very nearly given by

where GM is the metacentric height and k is obtained from the following definition :

The moment of inertia of a solid body about any axis is found by adding together the product of each weight making up the body and the square of its distance from the axis. (This is analogous to the moment of inertia of a plane area about an axis, dealt with in Chapter XVII.) If for a ship this axis is the axis of oscillation, W the weight and I the moment of inertia about the axis, then k is such a quantity that I = W × k2, and k is termed the radius of gyration.

The calculation for k is a most laborious one, but it has been done in a few cases, and having also the metacentric height, an estimate could then be made of the time of oscillation from the above formula. Practical agreement was found to exist between the actual and the estimated times of oscillation, even although the rolling could not have been unresisted.

The formula shows that to make the period long, i.e. to increase the time of oscillation, it is necessary to

(1) increase the radius of gyration, and, or
(2) decrease the metacentric height.

(The longer the period of a ship the more likely is she to be steady in a seaway.) Of these two, the first is of the lesser importance, because the distribution of the weights is governed by other features of the design than the desirability of obtaining a long period. The formula, however, shows clearly that winging weights, i.e. placing them at the sides, operates in the direction of increasing the period, although any practicable shift of weights on board a

war-ship can only have a small effect on the period. We should expect, therefore, to find that an armoured ship would roll more slowly than an unarmoured ship of about the same displacement and metacentric height, and this is confirmed by experience of ships in the Royal Navy.

A very considerable effect in lengthening the period is obtained by reducing the metacentric height. Thus in the Royal Sovereign, in which ship the period is 8 seconds and the GM about 3 ft., suppose the GM is reduced to 3 ft., without altering the radius of gyration. Then we should get a period of 864 seconds, or an increase of 8 per cent.

An interesting application of the above principles is found in the current practice of many merchant vessels. In many trades, voyages have to be undertaken with little or no cargo, because of the absence of return freights. It is necessary, for seaworthiness and proper immersion of the propellers, to sink the vessels by means of water-ballast. This has usually been placed in the doublebottom compartments. This, however, frequently pulls down the C.G. of the ship so far as to give the ship a large GM. This causes a very quick period, and in some cases this has not merely rendered the ship uncomfortable, but actually unsafe. In many ships, therefore, it is the practice to provide tanks in the 'tween decks and hold at the sides, and even on the upper deck. These tanks below are frequently large enough to hold ordinary cargo when necessary, but for "light" voyages they can be filled with water. The weight thus added, while giving sufficient immersion, does not produce excessive GM, and being at the sides tends to lengthen the period by increasing the radius of gyration.

The assumption used in obtaining the above formula for the period from side to side, viz. T = 0·55 is, that the righting

GM

lever varies directly as the angle or GZ = GM × 0, i.e. it assumes that the curve of stability is a straight line up to the angle considered. Under this condition large and small inclinations will be performed in the same time. A ship rolling in this manner is said to be isochronous.

Although the various assumptions made in obtaining the above formula are not strictly true, yet it is found by actual experiment that, within angles of 10° to 15° of the vertical, ships are very nearly isochronous in their rolling. This is the case although the ship experiences resistances which eventually bring her to rest.

The following are the approximate periods of some typical ships, i.e. the time from port to starboard, or vice versa.