OF A FLOATING BODY.

231

156. The period of oscillation will be large, and the ship will thus be steady among waves, if the metacentric height GM is made small.

On the other hand the stiffness of the ship under sail will be improved by increasing GZ and therefore also GM.

The stability of a ship therefore involves the two antagonistic qualities of steadiness and stiffness; and a compromise is effected by starting with a small metacentric height GM, and making the curve of statical stability rise rapidly, as shown in fig. 40, p. 161.

A steamer will recover the upright position when rolling among waves to any inclination short of the angle of vanishing stability; but a sailing ship, heeled steadily over by the press of sail, would capsize if inclined beyond the angle of maximum righting moment.

A squall which strikes the ship during a windward roll is dangerous, because it acts through a larger angle and imparts greater energy to the ship, and is thus more likely to carry the ship beyond this critical inclination.

157. Considering that GM is small, the motion of the ship may be imitated by supposing it, or a model, to oscillate like a compound pendulum about an axis of suspension through M, and G now oscillates on the arc of a small circle; but, strictly, K2 must be replaced by

K2+GM2.

The motion of the ship can also be imitated by a cradle, rocking on the curve of buoyancy BB2, which rolls on a horizontal plane.

A better imitation would be secured by supposing this horizontal plane smooth, so that G moves in a vertical

232 ANGULAR OSCILLATIONS OF A SHIP AGROUND.

line, and vertical oscillations come also into existence; these vertical oscillations can be allowed for by supposing the axis of suspension through M to be supported on springs; but in any case the complete solution of the oscillations of a ship leads into difficulties.

Treating the reaction of the water as acting hydrostatically in the case of the ship in § 142, aground along the keel and heeled over to a position of equilibrium, then for an additional small angular displacement 0, in which B2N', GH' denote the perpendiculars from B2, G on the new water plane, the additional righting moment, measured in fts of water multiplied into feet, due

(i.) to the wedges of immersion and emersion is

Aksin 0,

where Ak'2 denotes the moment of inertia, in ft*, of the plane of flotation about the axis through F"; (ii.) to the movement of B2 is - – V2. B2N. sin 0; (iii.) to the movement of G is - V. GH. sin 0.

2

Then V(K2+GK2) denoting the moment of inertia in quintic feet, ft5, about the keel, the oscillations of the ship will synchronize with a pendulum of length

so that the denominator must be positive for the equilibrium to be stable.

CHAPTER VI.

EQUILIBRIUM OF LIQUIDS IN A BENT TUBE. THE THERMOMETER, BAROMETER, AND SIPHON.

158. It has been proved in § 25 that "the common surface of two liquids which do not mix is a horizontal plane"; and in § 24 that "the separate parts of the free surface of a homogeneous liquid filling a number of communicating vessels all form part of one horizontal plane."

This last theorem no longer holds if two or more liquids of different densities, which do not mix, are poured into the communicating vessels.

To illustrate this difference, take a bent tube AB (fig. 50) and pour into the branches two different liquids, of densities σ and p, say mercury and water, or oil and water, so that the upper free surfaces stand at H and K and the plane surface of separation at the level AB.

Then if p denotes the pressure of the atmosphere, and h, k the vertical heights of H, K above AB, the pressure at A and B will be (§ 21) p+oh and p+pk; and these pressures being equal (§ 19)

which proves the

oh=pk, or h/k=p/o;

THEOREM.—“The vertical heights of the columns of two liquids above their common surface are inversely as the densities."

234

EQUILIBRIUM AND STABILITY

159. Suppose, for example, that the waters of the Mediterranean and the Dead Sea were in communication by a subterranean channel, reaching to a depth h below the surface of the Dead Sea, and k below the surface of the Mediterranean.

Then, according to the data of §§ 44, 76, if the waters of the two seas balance in this channel,

160. For the Stability of the Equilibrium of the liquids in the bent tube, it is requisite that the denser liquid should occupy the lower bend of the tube; otherwise the lighter liquid would be underneath the heavier liquid at A, and the equilibrium would be unstable, as shown in § 26.

The liquid in the bend AB may be replaced by any other liquid and the equilibrium will still subsist; but it will be unstable if the density of this liquid is less than the densities and p of the two other liquids.

σ

OF LIQUIDS IN A BENT TUBE.

235

Suppose then that initially a certain amount of liquid of the greater density is resting in a U-shaped tube, reaching to the same level in each branch.

If the lighter liquid, of density p, is now poured gradually into one of the branches, the equilibrium will be established in the bent tube when the heights h and k of the upper surfaces above the common surface are inversely as the densities σ and p.

But after a certain amount of the lighter liquid has been poured in, the denser liquid will be driven out of the bend; and now the lighter liquid where it is under the denser, will be in unstable equilibrium, and ultimately will bubble up to the upper surface, where it will form a separate column.

Provided the branches are vertical, or straight and of uniform section, the equilibrium in the other branch will be unaffected; otherwise a rearrangement takes place.

161. The equilibrium of the liquids as a whole is stable, supposing that a membrane or piston at the surface of separation prevents any instability of this surface.

For if the liquid column in the bent tube is displaced, so that H descends to H' and K rises to K', then taking for simplicity the tube as of uniform bore (fig. 50),

HH' = AA' = BB'=KK'=x, suppose;

and now if further motion is prevented by a stop valve s.v. in the bend of the tube, and the liquid in the bend AB is supposed of density p', the pressure on the side B of the valve exceeds the pressure on the side A by

pk+p'(c+x) —oh— p’(c—x)=2px,

where c denotes the height of AB above the stop valve; and therefore the liquid column tends to return to its original position when the valve is opened.