THICKNESS OF THE FILM.

421

320. In these investigations the weight of the liquid film has been taken as insensible; otherwise the weight would influence the result, as in tents and marquees.

But the thickness of the liquid films, as determined by Reinold and Rucker from optical measurements, may be as small as 12μμ; and half a litre of oil poured on the surface of the sea has been found to cover 108 cm2, or 100 m square, without losing its continuity, implying a thickness of 50μu; but this thin film is still effective for checking the ripples and small waves and calming the surface of the sea in a storm.

Many interesting maximum and minimum problems can be solved in a simple manner from the mechanical considerations involved in the theory of flexible surfaces under tension; as for instance :

The circle has the greatest area for given perimeter ; illustrated by an endless thread in a plane film, when the film in its interior is broken.

The sphere has the greatest volume for given surface; illustrated by the soap bubble.

Two segments on given bases and of given perimeter will enclose a maximum area when they are arcs of equal circles, realised by passing the endless thread through rings at the ends of the bases; and so also for spherical segments on given circles, realised by blowing bubbles on the ends of a frustum of a cone; etc,

Examples.

(1) Prove that the height of a flat drop of mercury is a mean proportional between the diameter of a capillary tube, and the depth to which mercury is depressed in it, supposing the angle of contact 180°.

(2) Investigate the coefficient of expansion of the radius of a soap bubble, supposing that the surface tension diminishes uniformly with the temperature. (3) A soap bubble of radius a is blown inside another of radius b; and the radii change to a' and b' when the atmospheric pressure changes from p to p'. Prove that

Ρ ́·b′ b (a2 — a′2)(b ́3 — a'3)+a3b ́3 — a′3b3

=

p′ ̄b b'(a2 — a'2)(b3 — a3)+a3b′3 — a′3b3°

(4) Prove that a flexible surface, of superficial density w lb/ft2, hanging as a horizontal cylinder the vertical cross section of which is a catenary, is changed by a pressure difference p lb/ft2 on its sides into a cylinder in which the tension across a generating line is still wy, where y is the height above a fixed horizontal plane, and in which the radius of curvature is changed to

if y=a where the slope =0; also that
(p+w cos )y=(p+w)a.

Determine the equation of this curve.

(5) Prove that if r, denote the radii of curvature of a pair of perpendicular normal sections of the surface in § 309, making an angle & with the lines of curvature; and if t, t, and u denote the corresponding normal tensions and tangential stress, due to a pressure difference p; then

where R, R' denote the principal radii of curvature.

CHAPTER X.

PRESSURE OF LIQUID IN MOVING VESSELS.

321. When a vessel containing liquid is moving steadily, as for instance a locomotive engine, with given acceleration a (ft/sec2), an attached plumb line is deviated from the vertical; and the surfaces of equal pressure and the free surface will be perpendicular to this plumb line, when the liquid is moving bodily with the vessel.

If the liquid fills the vessel completely so that there is no free surface, the liquid will move bodily with the vessel, provided the vessel has no rotation.

If however a vacant space is left, which may be supposed filled with some other liquid of a different density, oscillations will be set up in the free surface or surface of separation; but these oscillations die out rapidly in consequence of viscosity (§ 4), until the liquid and vessel move together bodily.

322. No oscillations however need be set up in the free surface by a vertical motion of the vessel (although Lord Rayleigh asserts that the horizontal free surface may become unstable), nor will the plumb line be deviated; this we may suppose realised in Atwood's machine, or else, initially, in the scales of a common balance, when equilibrium is destroyed.

424

PRESSURE IN ASCENDING

Suppose then that a bucket A and a counterpoise B, or else two buckets A and B, are suspended by a rope over a pulley, and that equilibrium is destroyed and motion takes place, in consequence of the inequality of the weights of A and B.

Denoting these weights in lb by W and W', by T pounds the tension of the rope, by a the vertical acceleration of A and B, and by g the acceleration of gravity, in ft/sec2; then by the principles of Elementary Dynamics and by Newton's Second Law of Motion,

We suppose the preponderating bucket A to be reduced to rest by applying to it an upward acceleration a; so that now the pressure at any depth z in the water in the bucket becomes changed from

If B was also a bucket of water, the pressure at a depth in it would be changed from

Dz to Dz1+

D2(1+2).

If the buckets are cylindrical and of weight negligible compared with the water they contain, then the hydrostatic thrust on the bottom of the buckets is

W(1-9) or W(1+9)

each equal to T, as is otherwise evident.

AND DESCEnding buCKETS.

425

Barometers attached to A and B, standing at a height h when at rest, would now have heights

So also with a bucket attached to a spring, performing vertical, simple harmonic oscillations, or placed on a vessel performing dipping oscillations; or with the water on the top of the piston of a vertical engine; a horizontal plane of cleavage may make its appearance when the amplitude and speed of the oscillations is sufficiently increased.

323. Suppose now that in each bucket a part of the weight, W or W', consists of a piece of cork of S.G. 8.

If the corks are floating freely no change will take place in consequence of the motion.

But if completely submerged by a thread attached to the bottom of the bucket, then denoting the tensions of the thread in A by Plb, and the weight of the cork by Mlb, the buoyancy of the cork at rest will be M/slb; and therefore in motion will be

and if s > 1, P becomes negative and the body must be supposed suspended by a thread from the top of the bucket.

For the tension of the thread in B the sign of a must be reversed.

Suppose WW', so that the buckets balance; then if the thread holding down the cork M in A is cut, the