458

THE FREE SURFACE

It is more convenient now to employ the absolute unit of force; so that the attraction of pure gravitation on a plummet weighing Wg is WG dynes, where G denotes the acceleration of gravity on a projectile or freely falling body; and the centrifugal force at a distance y cm from the polar axis is Wyw2 dynes, where w denotes the angular velocity of the Earth, in radians per second.

Producing the plumb line to meet the equator EQ in Q, then by the triangle of force EQP (fig. 100)

Now in the case of the surface of the Earth, G is so nearly constant and equal to g (980) and EP is so nearly equal to R the mean radius of the Earth, 10o÷π cm, that we may, as a close approximation, put

EQ=e2. MP, where e2= Rw2/g,

so that PQ is the normal to an ellipse of excentricity e passing through P; and the surface of the Ocean may be taken as the spheroid generated by the revolution of this ellipse about the polar axis.

This would be accurately true if the acceleration of pure gravity at a distance from the centre was G(r/R); and g at any point P would now vary as the normal PQ.

OF THE OCEAN.

The time of a sidereal revolution of the Earth being
T=23 h 56 m 4 s=86164 s,

459

358. Denoting the equatorial and polar semi-axes of the spheroidal surface by a and b, then (a-b)/a is called the ellipticity and denoted by e; and

€ = 1−√√/(1 − e2) ≈ 1e2.

The above value of e2 gives e=1/578; but geodetic measurements make e=1/300 about; the discrepancy is due to the fact that the solid Earth is not spherical or centrobaric, but that its shape partakes of the ellipticity of the Ocean, and to precisely the same amount; showing that the solid Earth was once in a molten viscous condition, during which time it took the present shape.

Suppose we increase w to , so that RQ2/g=1; then with the above value of e2, = 17w; and now the plumb line would be parallel to the Earth's axis, and would point to the Pole Star; at the equator water would fly off into space and bodies too, unless fastened down to the ground; and the water would collect in lakes with the free surface always parallel to the equator.

This implies however that the solid part of the Earth is rigid and does not change its spherical shape; but practically the solid form would be deformed into a spheroid, to a much greater extent than at present.

359. Plateau has devised an apparatus by which these phenomena may be imitated; a vertical axis is fixed in a vessel of water, and oil, of equal density, is placed on a solid nucleus fixed to the axis; the spheroidal form is closely imitated when the axis is revolved (fig. 101).

Here the constraining cause is the capillarity tension of the surface of separation, T dynes/cm suppose; and it can be proved as an exercise that if the mean radius is R cm and the density σ g/cm3, the ellipticity due to a small angular velocity w is ow2R3/T.

360. According to astronomical definitions the angle PGQ is called the latitude of the place; not the angle PEQ, which is distinguished as the geocentric latitude, the angle EPG being called the angle of the centre; the tangent plane at P perpendicular to the plumb line GP is called the sensible horizon, and the parallel plane through the centre E the rational horizon of P.

The angle of the centre EPG is the gradient of the free level surface with respect to the mean spherical surface through P; denoting it by p, and the geocentric latitude by 0,

sin =(EG/EP)sin 0=e2cos 0 sin 0 e sin 20.

Thus in latitude 45°, where p is greatest, a river flowing south is running away from the centre of the Earth on an apparent gradient of about one in 300.

The Mississippi rises in latitude 75°, and flows nearly due south into the sea in latitude 30°, a distance of 900 geographical miles, at an average gradient with respect to the Earth's centre of one in 320; so that if the source is one-quarter of a mile above sea level, the mouth will be about 2 miles farther from the centre of the Earth.

CHAPTER XI.

HYDRAULICS.

361. The word Hydraulics means primarily the science of the Motion of Water in Pipes; but it is now extended to cover the elementary parts of the practical science of the Motion of Fluids.

This includes the Discharge from Orifices, the Theory of Hydraulic Machinery, such as Water Wheels, Turbines, Paddle Wheels and Screw Propellers, Injectors, etc., which can be treated by the aid of Torricelli's and Bernoulli's Theorems; and the Motion in Pipes, Canals, and Rivers, taking into account the effect of Fluid Friction so far as it can be treated in an elementary manner.

362. Torricelli's Theorem.

The velocity v of discharge of water from a small orifice a depth h below the free surface was given by Torricelli (1643) as the velocity v acquired in falling from the level of the free surface, so that

v2=gh, or v=√(2gh);

and v is then called the velocity due to the head h.

This is argued by asserting that the hydrostatic energy of the water, Dh ft-lb per ft3, or h ft-lb per lb, becomes converted on opening the orifice into the kinetic energy Dv2/g ft-lb/ft3, or v2/g ft-lb/lb.

462

TORRICELLI'S THEOREM ON

Thus the jet of water, if directed nearly vertically upwards, would nearly reach the level of the free surface; and if directed in any other direction will form a parabolic jet, of which the directrix lies in the free surface of the still liquid.

The cross section of the jet OVR, while continuous and not shattered into drops, will be inversely as the velocity; and the horizontal component of the velocity being constant, equidistant vertical planes will intercept equal quantities of water, so that G the C.G. of the water will coincide with the c.G. of the parabolic area cut off by the chord; and the height of the c.G. of the jet cut off by a horizontal chord OR will be two-thirds of the height of the vertex (fig. 104, p. 469).

If the jet could be instantaneously reduced to rest and frozen, it could stand as an arch, without shearing stress across normal sections.

If the vessel is in motion, the velocity of efflux v is still taken as due to the head of the pressure p; in this way the efflux from an orifice in a rotating vessel (Barker's Mill or a Turbine) is given (§ 345) by v= √(2gz+y2w2), or from an orifice in an ascending or descending bucket; balanced by a counterpoise at the end of a rope over a pulley by v = {2(g±a)z} (§322); the student may work out the motion of the buckets completely as an exercise.

363. The velocity of efflux v must be reckoned not exactly at the orifice, but a little in front at the point where the jet is seen to contract to its smallest cross section; this part is called the vena contracta, and the ratio of the cross section of the vena contracta to that of the orifice is called the coefficient of contraction, and denoted by c1.