Page:The American Cyclopædia (1879) Volume IX.djvu/131

HYDROMECHANICS 123 as much as one of one square foot. The sec- ond law is not so evident, but will be made clear by considering that with a given surface, when the velocity is doubled, twice the quan- tity of liquid will move through twice the space in the same time, and will therefore, ac- cording to the principles of mechanics, have a fourfold momentum. The resistance, there- fore, offered to a plane surface moving at right angles against a liquid, is measured by the area of the surface multiplied into the square of the velocity. It has been found that a square foot surface, moved through water with a velocity of 32 ft. per second, meets with a resistance equal to a weight of 1,000 Ibs. When the motion of a body in a liquid is very slow, say less than 4 in. per sec- ond, depending on the size of the body, the larger body requiring to move more slowly, the above laws are not rigidly followed, but the resistance is divided into two components, one of which is proportional to the simple ve- locity, and the other to the square of the ve- locity. The most accurate results in experi- menting with slow motions were obtained by Coulomb, who used his torsion balance. One of the most interesting problems in mathemat- ics has been to determine the form of a solid which will meet with the least resistance in moving through water. This form is called the " solid of least resistance," and is approached as near as practicable in the construction of ships. Theory of Waiie in Liquids. "When a pebble is dropped into still water, a series of circular waves is formed upon its surface, which extend themselves from the centre in all directions. These waves consist of alter- nate elevations and depressions, which have the appearance of following one another in the direction of the radii of the circle. It is how- ever only an appearance, as may be readily proved by throwing a cork upon the undu- lating surface, when it will be observed only to rise and fall, and the undulations will ap- pear to glide beneath it. The wave then is an oscillation of the liquid upward and down- ward, and the force which causes it is gravity. The pebble when it strikes the water displaces a portion, which rises on every side to a cer- tain height, and then, its momentum being lost, and being higher than any portion of liquid around it, it falls ; but the momentum it has acquired carries it below the level, and an exterior ring is forced upward, which in descending also produces a successor; and thus a series of circular waves is formed of gradually diminished height but of increased diameter, until, at a very great distance in calm water, the force of the primary impulse is lost. When two waves proceeding from dif- ferent centres meet one another in such a way that the elevations coincide, a united wave will be produced having a height equal to that of its two components, and a depression equal to that of the other two; but if the elevation of one corresponds to the depression of the other, Fifl. 21. the resulting elevation and depression will be equal to the difference of elevation and depres- sion respectively of the original waves. If they are equal, the result will be the oblitera- tion of both. This phenomenon is called the interference of waves. It is susceptible of de- monstration that the undulations of waves are performed in the same time as the oscillations of a pendulum whose length is equal to the dis- tance between two eminences, or the technical breadth of the wave. Form of Surface of Rota- ting Liquid. From the principle of the equilib- rium of fluids, that the surface of the liquid at rest must be a level which is perpendicular to the direction of the force of gravity, it fol- lows that when two or more forces act upon a liquid to change the position of its sur- face, the resultant of these forces will be perpendicular to the surface. Therefore, if a cylindrical or conical vessel, fig. 21, containing a liquid, is rotated on its axis A B, all the particles on the sur- face will be acted upon by two forces, that of gravity, in a vertical direction represented by A C or C E, and the centrifugal force, repre- sented by C D or E F, which is horizontal, and varies in intensity with the distance of the particles from the axis or centre of motion. The surface of the liquid will therefore be de- pressed in the middle, and will be at every point perpendicular to the resultants A D, C F, &c., which will therefore be normals; and it may be demonstrated that the subnormals A C, C E, &c., are equal, and therefore that the surface of the liquid is a paraboloid. A Level Surface. Let it be assumed that if the earth were entirely covered with water, and at rest, with no force acting upon the water except gravity, it would have the fqrm of a perfect sphere. But it has been found to have the form of an oblate spheroid, the ratio of its polar to its equatorial diameter being about 299 to 300. Its oblate form is caused by its rotation on its axis. Let abed, fig. 22, be the section of a liquid sphere, pass- ing through its axis of rotation a o, and let / he any point on its surface. The revolution of the sphere on its axis will generate a cen- trifugal force in the direction of/e, par- allel to the plane of the equator c d, and perpendicular to the axis a J. Now, if/ h repre- sent the force of gravity andfe the centrifugal force, fg will represent the resultant of these two forces, and the surface of the liquid, being free to move, must become perpendicular to this resultant at every point. The surface of a Fiu. 22.