A body. The ends of this axis are supported; and the body being drawn away from its position of equilibrium and then left free will move to and fro. Now it is found by theory that the motion of this real pendulum is exactly the same as that of a simple pendulum of some definite length which can be calculated when the form and the substance of the body are known; this length is called the length of the equivalent simple pendulum. If we measure along the line through C and the centre of gravity of the body, a distance CO equal in length to the length of the equivalent simple pendulum, then O is called the centre of oscillation, while C is called the centre of suspension. The centre of gravity of the body will be at some point between C and 0. B 339. It is remarkable that the centres of oscillation and suspension are convertible; this means that if the body instead of turning round the horizontal axis at C turns round a parallel axis at O, then C becomes the new centre of oscillation. 340. The position of the centre of oscillation can be determined as we have said by theory; but it may also be found by experiment. For example if a slender rod oscillate about an axis through one end at right angles to the rod it is found that it oscillates in the same time as a simple pendulum two thirds of the length of the rod. Thus the centre of oscillation is distant two thirds of the length of the rod from the fixed rod. The statement can be verified by making the rod oscillate about an axis through the point thus assigned; then by Art. 339 the time of oscillation will be the same as before. 341. The rule found by theory for the length of the equivalent simple pendulum in the case of any body is the following. Suppose the body to consist of any number of equal small particles, then the required length is a fraction to be calculated thus: The numerator is the sum of the squares of the distances of the particles from the horizontal axis; the denominator is the sum of the distances of the particles below the horizontal plane through the axis from that plane, diminished by the sum of the distances of those above, when the body is in its lowest position. 342. The centre of oscillation does not necessarily fall within the body. It is obvious from the diagram of Art. 338 that the parts of the body above and below Crespectively are always tending by their weights to move the pendulum in contrary directions, so that if these two parts are so adjusted as to produce nearly equal effects the motion may be very slow indeed, and thus CO may be very long and consequently the point O quite beyond the body. Musicians use a small pendulum called a metronome for the sake of marking time; though very short it can be made to oscillate in a second or even in a longer time. is of the form represented in the diagram, namely a rod with balls at the ends. The upper ball can be moved to any position which may be desired, and held fixed in that position by a screw; thus the metronome can be made to Oscillate at a quicker or slower rate as may be required. It 343. It will be observed that a heavy body oscillates in the same manner as if the whole weight were collected at the centre of oscillation, not as if it were collected at the centre of gravity. It is sometimes stated incautiously that the weight of a body may always be supposed to be collected at the centre of gravity; but the present case shews that such a statement is too wide: see Art. 169. 344. The following very important result is demonstrated by theory with respect to the motion of any body. The motion of the centre of gravity of a body is exactly the same as the motion of a particle having a mass equal to the mass of the body and acted on by forces equal and parallel to those which act on the body. The reader will scarcely be prepared to understand completely this very remarkable statement, but even an imperfect notion of it will be of service. Take, as a simple example, a top spin ning and moving on the ground. There are various forces acting, the weight of the top, the resistance and the friction from the ground, and the resistance of the air. Suppose all these forces moved up to the centre of gravity of the body, each force remaining parallel to its original direction, and let their resultant be found; then if this resultant act on a particle of the same mass as the whole top the motion will be the same as the actual motion of the centre of gravity of the top. 345. Another result of the same kind is the following. The motion of a body round its centre of gravity is the same as the body would take if its centre of gravity were fixed and the body were left to turn round under the influence of the forces really acting. But this, like the former proposition, is beyond the range of an elementary work. XXV. FLUIDS. 346. Two opposing principles are found to operate extensively throughout the material world; one is the principle of cohesion which tends to bind the component particles of bodies together, aud the other is the principle of repulsion which tends to separate the particles. The principle of cohesion is perhaps connected with that of attraction between bodies at a distance; the principle of repulsion is perhaps identical with heat, or at least intimately connected with it. Now the three forms under which matter presents itself depend upon the relative influence of these two principles. In solid bodies cohesion prevails over repulsion, so that the particles form one connected mass, not to be separated without the application of force. In air and gases the principle of repulsion predominates, and the particles require the application of force in order to keep them in contact or near each other. The third form of matter, namely that of water and other liquids, is one in which neither of the two principles is predominant; the particles can be separated by the application of forces so slight as to be practically insensible, but they do not require to be confined in every direction, like those of air and gases to prevent them from escaping. 347. The term fluid includes two classes of objects, namely liquids like water, and gaseous bodies like air; the two classes have some properties in common, and each class has also some of a special kind. We shall treat first of liquids and then of gases. The most common liquid is water, and this may be taken as the type of all. Hence the science which we are about to consider has received names derived from the Greek word for water. The term Hydrostatics has been applied to all that concerns the mechanical properties of liquids in equilibrium, and Hydrodynamics to the subject of liquids in motion: the term Hydraulics is sometimes applied to the theory of machines which depend on the action of liquids. 348. The general properties which we are about to consider are those which belong to what are called perfect liquids. By perfect liquids we mean such as offer no resistance whatever to the separation of their parts, and on this account adapt themselves to the shape of the vessels containing them. Strictly speaking no liquids are perfect; but still for water and other liquids there will be no error of practical importance introduced in the statements we shall make. There are however substances which though liquids are far from being perfect in the sense we have explained, and to which therefore our subsequent statements will not apply; for instance tar or melted glue. Water approaches more nearly than oil to the idea of a perfect liquid, and alcohol more nearly than water. 349. It was formerly supposed that liquids were incompressible; that is to say it was held that a liquid could not have its bulk diminished by any pressure however great. An experiment was made at Florence, and thence known as the Florentine experiment, which seemed to confirm this notion. Water was enclosed in a hollow globe of gold; the globe was squeezed in such a manner as to alter its form, and therefore by the conclusions of Geometry to diminish its size, and it was found that the water was forced through the pores of the gold. But it is now well ascertained that water is compressible, though the compressing force must be very great in order to produce a sensible effect. The standard fact may be put in the following form, which will be fully comprehended as the reader proceeds with the subject: water when pressed by a column of water 33 feet high has its density increased by 000046 of its original density. Also the increase of density will be in proportion to the pressure; so that under the pressure of a column of water 3300 feet high the density would be increased by 0046 of the original density, and under the pressure of a column of water 7000 feet the density would be increased by about 01 of the original density. Or we may put the last fact in this form at the depth of 7000 feet in the sea a mass of water will lose of the bulk it would have at the 1 100 |