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 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 and p.

σ

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.

236

OSCILLATIONS OF THE LIQUID COLUMN.

The column will now oscillate; and, as in § 148, the force on the column tending to bring it back to its position of equilibrium when displaced through a distance x being 2p'wx, where a denotes the uniform cross section of the tube, the column will oscillate like a pendulum of length

W

1=2pw

where W denotes the weight of the liquids, and a the length of the filament in the bend AB.

(Principia, lib. II., Prop. XLIV.) Suppose, however, that the branches of the tube are not vertical, but curved, so that the inclinations to the vertical at the points A, B, H, K are a, ẞ, 0, p.

Then to push the column through a small distance x from its position of equilibrium by a piston at H will require a thrust reaching from zero to

pw(x cos +k-x cos ẞ) + p'w(x cos ẞ+x cos a) - ow(x cos a+h− x cos 0)

= {p cos +(p'-p)cos ẞ+(p'-σ)cos ato cos 0}wx, so that, if the piston is removed, the column will oscillate like a pendulum of length

l=

W/w

ρ cos +(p'-p)cos ẞ+(p'−σ)cos ɑ+σ cos reducing for a homogeneous filament, of length c, to

Similarly for the oscillations of any number of liquids in a uniform bent tube; but if the bore of the tube changes, as in a marine barometer, the problem is complicated by the variations of velocity in the tube.

HARES HYDROMETER.

237

162. Hare's Hydrometer. This is an application of the principle of the Theorem of § 158; it consists of two vertical glass tubes AH and BK, dipping into vessels at A and B, containing two liquids whose densities, σ and p, are to be compared (fig. 51). The upper ends of the tubes are cemented into a receptacle C, from which the air can be partially exhausted by an air pump or other means; and the liquids now rise in AH and BK to heights h and k above their level at A and B, which heights are inversely as the densities, or such that

oh=pk, or p/σ=h/k.

An apparent tension draws up the liquid columns in the tubes; for this reason any small pressure below the atmospheric pressure is sometimes called a tension, because the difference between this pressure and that of the atmosphere is a negative pressure, or a tension thus it is usual to speak of the tension of aqueous and other vapours; but the word tension is sometimes improperly applied to very high pressures, such as those due to the gases of fired gunpowder.

Examples.

(1) Two equal vertical cylinders of height 7 stand side by side and there is free communication between their bases. Quantities of two liquids of densities P1, Pg which would fill lengths a and c respectively of the cylinder, are poured in, and rest in stable equilibrium, each liquid being continuous.

A given quantity of a liquid of density p2, intermediate between p1 and pg, is poured slowly into one of the cylinders.

P1