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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, W-T T-W' W-W'

a

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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

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

Dz to Dz(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

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each equal to T, as is otherwise evident.

AND DESCENDING BUCKETS.

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

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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 P lb, and the weight of the cork by Mlb, the buoyancy of the cork at rest will be M/s lb; and therefore in motion will be

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

426

IMPULSIVE PRESSURE

equilibrium is destroyed; and the student may prove as an exercise that the bucket A will descend, and the cork will rise through the water, with accelerations respectively

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This treatment, as in § 148, ignores the motion of the water due to the passage of the cork; but, as in § 149, the result can be corrected for a small spherical or cylindrical cork in a large vessel.

324. If the bucket A strikes the ground with velocity v and is suddenly reduced to rest, an impulsive pressure is set up for an instant in the water.

Suppose, however, that the impact takes an appreciable time, t seconds.

To stop the body A weighing W lb, moving with velocity v ft/sec, in a short time t seconds, requires an average resistance R of the ground, given in pounds by

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The product Rt of the force of R pounds and of the t seconds for which it acts is called the impulse, in second-pounds; and its mechanical equivalent Wv/g is called the momentum, also in sec-lb, of W lb moving with velocity v ft/sec.

To stop the water in the bucket reaching to a depth z ft requires therefore a force Dzav/gt pounds, a ft2 denoting the cross section of the bucket, or a pressure Dzv/gt lb/ft2, compared with which the pressure due to gravity is insensible when t becomes small, so that the bucket runs the risk of bursting when it strikes the

IN MOVING LIQUID.

427

ground, a tension Drzv/gt pounds per unit depth being set up in the circumferential hoops, if of radius r.

As for the bucket B, the rope becoming slack, it moves freely under gravity, and the pressure of the water in it is reduced throughout to zero, or atmospheric pressure.

325. When a closed vessel, completely filled with water, and moving bodily in a given direction with velocity v, is suddenly stopped, the water comes to rest simultaneously; so that the impulse on any portion of the water is equal and opposite to the momentum of this liquid.

Suppose this portion of water is removed and replaced by an equal solid, of S.G. 8 and weight M lb; the momentum of this solid, moving bodily with the liquid with momentum Mv/g sec-lb, is changed by the impulse Mv/gs of the surrounding liquid into

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in the direction opposite to the original motion; so that if the body is lighter than the surrounding water, the body will recoil with this momentum.

If the body is like the cork attached by a thread to the bottom of the bucket A, the impulsive tension of the thread will be

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The impulsive pressure at depth z in the water on impact of the bucket is therefore (in sec-lb/ft2)

Dzv/g;

or, if the velocity is suddenly reduced from v to v', the impulsive pressure is

Dz(v-v')/g.

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