118 HYDROMECHANICS FIG. 4. and standing vertically, a pressure may be produced on its bottom several thousand times that due to the weight of the water alone. In accordance with this law of hydrostatic pressure, a liquid will rise to the same height in different branches of the same vessel, wheth- er these branches be great or small. Thus, water con- tained in the U- shaped vessel, fig. 4, will rise to the same height in both branches, which is an illustration of the principle that the pressure of a column of liquid is in pro- portion to its height and not to its quantity. This principle, how- ever, if it is entitled to such a name, proceeds directly from the principle of Archimedes that each particle in a liquid at the same depth receives an equal pressure in all directions. If however one leg of a U-shaped tube con- tain mercury and the other water, the col- umn of water will stand 13^ times as high as that of mer- cury. It follows from the fact that a liquid presses equally upon equal areas of a con- FIG. 5. taining vessel at the same depth, that if a hole is made in one side of a vessel, less pressure will be exerted in the direction of that side ; and therefore if the vessel is floated on water, as in fig. 5, it will be propelled in the direction of the arrow. Barker's centrifugal mill, a small model of which is shown in fig. 6, acts upon the same principle of inequality of pressure on opposite sides. The propelling force has been ascribed to the action of the escaping liquid press- ing against the atmos- phere, by which a cor- responding reaction is obtained ; but if the machine is placed in a vacuum, it will ro- tate with greater ve- locity than in the open air, which proves that the propelling force is the preponderance of pressure in one direction. The two following are important laws of hy- drostatics : 1. The hydrostatic pressure against equal areas of the lateral surfaces of cylindri- cal or prismoid vessels, commencing from the surface of the liquid, varies as the odd num- Fta. 6. Barker's Mill. -3 3 4- -..- 7
9- -
bers 1, 3, 5, 7, &c. 2. The hydrostatic pres- sure against the entire lateral surfaces of cylin- drical or prismoidal vessels is proportional to the square of the depth. The first law is de- monstrated as follows : Hydrostatic pressure in any direction at any point in a liquid is in proportion to the depth, a result due to the action of gravity ; therefore the mean pressure against any rectangular lateral area will bo on a horizontal line midway between the up- per and lower sides of such area. The depth of this line, proceeding from the surface of the liquid downward, varies as the odd numbers 1, 3, 5, 7, &c., as will be seen by an inspection of the adjoining diagram, fig. 7. The figures placed upon the dotted lines in the centre of the areas indicate the pressures upon those lines, and also the propor- tional pressures against those areas. The figures on the right side of the diagram in- dicate the pressures at points of equal vertical distances, while those upon the left in- dicate the total lateral pres- sures, which it will be ob- served are the squares of the number of areas included ; by which is demonstrated the second law, that the total lateral pressure against rectangular areas is in pro- portion to the square of the depth. The weight of a cubic foot of water is 62-5 Ibs. ; therefore the lateral pressure against a surface of a square foot, whose upper side is in the surface of the liquid, is 31-25 Ibs. From this it is easy to ascertain the pressure against a square foot, or any area, at any depth below the surface. Simply multiplying the number of feet below the surface by 2 and subtracting 1, multiplying the remainder by 31-25 and this product by the number of horizontal feet, will give the pressure of a stratum of water a foot deep, at any depth below the surface and of any length. To ascertain the entire pressure against the sides of a vertical cylindrical or prismoidal vessel, square the depth of the liquid in feet or inches, and multiply this by the lat- eral pressure against an upper vertical square foot or inch, as the case may be, remembering that the weight of a cubic inch of water is 5792 of an ounce, and therefore that the pres- sure against an upper lateral side is -2896 of an ounce. The total pressure exerted against the sides of a cylindrical pipe 60 ft. high and 2 in. in diameter is found as follows: 60" x 31-25 = 112,300. The diameter of the pipe being 2 in., the circumference of the inner surface is 2 x 3-141592 (the constant ratio) = 6-283184 in., or A- t yK of a foot. Therefore, 1 12,500 x A-i^L&A 58,904-92 Ibs. or 29-95 tons. The lateral pressure on the lower foot would be (60 x 2)- 1 = 119 x 31-25 x 'lAfyjiA = 1,959-64
1C 25 -7 8 -9 10 FIG. 7.
