365

BUOYANCY OF PONTOONS.

GENERAL SCHOLIUM.

245. THE principles established in art. 238 have an interesting application to military men, in the use of pontoons, and the buoyancy by which they become serviceable in the construction of temporary bridges. When the dimensions, magnitude, and weight of a pontoon are known, that weight can readily be deducted from the weight of an equal bulk of water, and the remainder is evidently the weight which the pontoon will carry before it will sink.

Pontoons as usually constructed, are prisms whose vertical sections are equal trapezoids, as exhibited in the marginal figure.

Suppose AB = L

CD = 1

AI= KB =

=

(L — 1) = 8 CI= D.

K B

G H

M

D

Uniform width of the pontoon =b: all in feet and parts. Suppose also CL: d, depth of the part immersed; w = weight in avoirdupois pounds of the water displaced; and c = 624lbs. weight of a cubic foot of rain water. Then, by the following expressions, which are left for the student to investigate, d may be found when w and the rest are given, and w may be found when d and the rest are given; also the maximum value of w.

Ex. 1. Given AB 21 feet, CD = 17 feet, CI = 2 feet, b = 42 feet. Required the weight of the pontoon and its load, when it is immersed to the depth CL of 1 feet. Ans. 8287 lbs. nearly.

Ex. 2. Suppose the weight of such a pontoon to be 900lbs. what is the greatest weight it will carry? Ans. 12014 lbs. Ex. 3. Suppose the weight of the above pontoon and its load to be 6000lbs. how deep will it sink in water? Ans. 1·1084 f = 13.3 inches.

HYDRAULICS OR HYDRODYNAMICS.

246. HYDRAULICS or HYDRODYNAMICS is that part of mechanical science which relates to the motion of fluids, and the forces with which they act upon bodies against which they strike, or which move in them.

This is a very extensive subject; but we shall here give only a few elementary propositions.

247. PROP. If a fluid run through a canal or river, or pipe of various widths, always filling it; the velocity of the fluid in different parts of it, AB, CD, will be reciprocally as the transverse sections in those parts.

That is, veloc. at A: veloc. at C: CD: AB; where AB and CD denote, not the diameters at A and B, but the areas or sections there.

B

D

For, as the channel is always equally full, the quantity of water running through AB is equal to the quantity running through CD, in the same time; that is, the column through AB is equal to the column through CD, in the same time; or AB × length of its column= CD x length of its column; therefore AB: CD :: length of column through CD length of column through AB. But the uniform velocity of the water, is as the space run over, or length of the columns; therefore AB: CD:: velocity through CD: velocity through AB.

248. Corol. Hence, by observing the velocity at any place AB, the quantity of water discharged in a second, or any other time, will be found, namely, by multiplying the section AB by the velocity there.

But if the channel be not a close pipe or tunnel, kept always full, but an open canal or river; then the velocity in all parts of the section will not be the same, because the velocity towards the bottom and sides will be diminished by the friction against the bed or channel; and therefore a medium among the three ought to be taken. So if the

velocity at the top be

that at the bottom

and that at the sides

82 feet per minute,

68

60

3)210 sum;

dividing their sum by 3, gives 70 for the mean velocity, which is to be multiplied by the section, to give the quantity discharged in a minute: and in many cases still greater accuracy will be necessary in determining the mean.

249. PROP. The velocity with which a fluid runs out by a hole in the bottom or side of a vessel, is equal to that which is generated by gravity through the height of the water above the hole; that is, the velocity of a heavy body acquired by falling freely through the height AB.

The momenta, or quantities of motion, generated in two given bodies, by the same force, acting during the same or an equal time, are equal. And the force in this case, is the weight of the superincumbent column of the fluid over the hole. Let then the one body to be moved, be that column itself, expressed by ah, where a denotes the altitude AB, and h the area of the hole; and the other body is the column of the fluid that runs out uniformly in one second suppose, with the middle or medium velocity of that interval of time, which is ¿hv, if v be the whole velocity required. Then the mass hv, with the velocity v, gives the quantity of motion hv ☀ v, or hv2, generated in one second, in the spouting water also g, or 32 feet, is the velocity generated in the mass ah, during the same interval of one second; consequently ah × g, or ahg, is the motion generated in the column ah in the same time of one second. But as these two momenta must be equal, this gives hv2 =ahg: hence then v2 = 2ag, and v = √2ag, for the value of the velocity sought: which therefore is exactly the same as the velocity generated by the gravity falling through the space a, or the whole height of the fluid.

2

250. For example, if the fluid were air, of the whole height of the atmosphere, supposed uniform, which is about 5 miles, or 27720 feet =a. Then 2ag: = (27720 × 16111⁄2) = 1335 feet = v the velocity, that is, the velocity with which common air would rush into a vacuum.

251. Corol. 1. The velocity, and quantity run out, at different depths, are as the square roots of the depths. For the velocity acquired in falling through AB, is as AB.

252. Corol. 2. The fluid spouts out with the same velocity, whether it be downward or upward, or sideways; because the pressure of fluids is the same in all directions, at the same depth. And therefore, if an adjutage be turned upward, the jet will ascend to the height of the surface of the water in the vessel. And this is confirmed by experience, by which it is found that jets really ascend nearly to the height of the reservoir, abating a small quantity only, for the friction against the sides, and some resistance from the air and from the oblique motion of the fluid in the hole.

253. Corol. 3. The quantity run out in any time, is equal to a column or prism, whose base is the area of the hole, and its length the space described in that time by the velocity acquired by falling through the altitude of the fluid. And the quantity is the same, whatever be the figure of the orifice, if it is of the

same area.

Therefore, if a denote the altitude of the fluid,

and h the area of the orifice,

also g = 16 feet, or 193 inches;

then 2h ag will be the quantity of water discharged in a second of time; or nearly 8h a cubic feet, when a and h are taken in feet.

So, for example, if the height a be 25 inches, and the orifice h = 1 square inch; then 2h√ag = 2√25 × 193 139 cubic inches, which is the quan

tity that would be discharged per second.

SCHOLIUM.

254. When the orifice is in the side of the vessel, then the velocity is different in the different parts of the hole, being less in the upper parts of it than in the lower. However, when the hole is but small, the difference is inconsiderable, and the altitude may be estimated from the centre to obtain the mean velocity. But when the orifice is pretty large, then the mean velocity is to be more accurately computed by other principles, given in the next proposition.

255. It is not to be expected that experiments, as to the quantity of water run out, will exactly agree with this theory, both on account of the resistance of the air, the resistance of the water against the sides of the orifice, and the oblique motion of the particles of the water in entering it. For, it is not merely the particles situated immediately in the column over the hole, which enter it and issue forth, as if that column only were in motion: but also particles from all the surrounding parts of the fluid, which is in a commotion quite around; and the particles thus entering the hole in all directions, strike against each other, and impede one another's motion: from which it happens, that it is the particles in the centre of the hole only that issue out with the whole velocity due to the entire height of the fluid, while the other particles towards the sides of the orifices pass out with decreased velocities; and hence the medium velocity through the orifice, is somewhat less than that of a single body only, urged with the same pressure of the superincumbent column of the fluid. And experiments on the quantity of water discharged through apertures, show that the quantity must be diminished, by those causes, rather more than the fourth part, when the orifice is small, or such as to make the mean velocity nearly equal to that in a body falling through the height of the fluid above the orifice. If the velocity be taken as that due to the whole altitude above the orifice, then instead of the area of the orifice, the area of the contracted vein at a small distance from it must be taken. See Gregory's Mechanics and Bossut's Hydrodynamique.

256. Experiments have also been made on the extent to which the spout of water ranges on a horizontal plane, and compared with the theory, by calculating it as a projectile discharged with the velocity acquired by descending through the height of the fluid. For, when the

aperture is in the side of the vessel, the fluid spouts out horizontally with a uniform velocity, which, combined with the perpendicular velocity from the action of gravity, causes the jet to form the curve of a parabola. Then the distances to which the jet will spout on the

H

horizontal plane BG, will be as the roots of the rectangles of the segments AC. CB, AD. DB, AE. EB. For the spaces BF, BG, are as the times and horizontal velocities; but the velocity is as

AC; and the time of the fall,

CB; therefore the distance BF

which is the same as the time of moving, is as is as AC. CB; and the distance BG as/AD. DB. And hence, if two holes are made equidistant from the top and bottom, they will project the water to the same distance; for if AC = EB, then the rectangle AC. CB is equal the rectangle AE. EB: which makes BF the same for both. Or, if on the diameter AB a semicircle be described; then, because the squares of the ordinates CH, DI, EK are equal to the rectangles AC. CB, &c.; therefore the distances BF, BG are as the ordinates CH, DI. And hence also it follows, that the projection from the middle point D will be farthest, for DI is the greatest ordinate.

These are the proportions of the distances; but for the absolute distances, it will be thus. The velocity through any hole C, is such as will carry the water horizontally through a space equal to 2AC in the time of falling through AC: but, after quitting the hole, it describes a parabola, and comes to F in the time a body will fall through CB; and to find this distance, since the times are as the roots of the spaces, therefore AC:CB: 2AC: 2 VAC. CB = 2CH = BF, the space ranged on the horizontal plane. And the greatest range BG= 2DI, or 2AD, or equal to AB.

And as these ranges answer very nearly to the experiments, this confirms the theory, as to the velocity assigned.

257. PROP. If a notch or slit EH in form of a parallelogram, be cut in the side of a vessel, full of water, AD; the quantity of water flowing through it will be of the quantity flowing through an equal orifice, placed at the whole depth EG, or at the base GH, in the same time; it being supposed that the vessel is always kept full.

EF

A

C

For the velocity at GH is to the velocity at IL, as VEG to EI; that is, as GH or IL to IK, the ordinate of a parabola EKH, whose axis is EG. Therefore the sum of the velocities at all the points I, is to as many times the velocity at G, as the sum of all the ordinates IK, to the sum of all the IL's; namely, as the area of the parabola EGH, is to the area EGHF; that is, the quantity running through the notch EH, is to the quantity running through an equal horizontal area placed at GH, as EGHKE, to EGHF, or as 2 to 3; the area of a parabola being of its circumscribing parallelogram.

B

Corol. 1. The mean velocity of the water in the notch, is equal to 3 of that at GH.

Corol. 2. The quantity flowing through the hole IGHL, is to that which would flow through an equal orifice placed as low as GH, as the parabolic frustum IGHK, is to the rectangle IGHL. This appears from the demonstration.

ON PNEUMATICS.

258. PNEUMATICS is the science which treats of the properties of air, or elastic fluids.

259. PROP. Air is a fluid body; which surrounds the earth, and gravitates on all parts of its surface.

These properties of air are proved by experience. That it is a fluid, is evident from its easily yielding to any the least force impressed on it, without making a sensible resistance.

But when it is moved briskly, by any means, as by a fan or a pair of bellows; or when any body is moved very briskly through it; in these cases we become sensible of it as a body, by the resistance it makes in such motions, and also by its impelling or blowing away any light substances. So that, being capable of resisting, or moving other bodies, by its impulse, it must itself be a body, and be heavy, like all other bodies, in proportion to the matter it contains; and therefore it will press on all bodies that are placed under it.

Also, as it is a fluid, it spreads itself all over on the earth; and, like other fluids, it gravitates and presses every where on the earth's surface.

260. The gravity and pressure of the air are also evident from many experiments. Thus, for instance, if water, or quicksilver, be poured into the tube ACE, and the air be suffered to press on it, in both ends of the tube, the fluid will rest at the same height in both legs. but if the air be drawn out of one end as E, by any means; then the air pressing on the other end A, will press down the fluid in this leg at B, and raise it up in the other to D, as much higher than at B, as the pressure of the air is equal to. From which it appears, not only that the air does really press, but also how much the intensity of that pressure is equal to. And this is the principle of the barometer.

C

261. PROP. The air is also an elastic fluid, being condensible and expansible : and the law it observes is this, that its density and elasticity are proportional to the force or weight which compresses it.

This property of the air is proved by many experiments. Thus, if the handle of a syringe be pushed inward, it will condense the inclosed air into less space, thereby showing its condensibility. But the included air, thus condensed, is felt to act strongly against the hand, resisting the force compressing it more and more; and, on withdrawing the hand, the handle is pushed back again to where it was at first. Which shows that the air is elastic.

262. Again, fill a strong bottle half full of water; then insert a small glass tube into it, putting its lower end down near to the bottom, and cementing it very close round the mouth of the bottle. Then, if air be strongly injected B b

VOL. II.