1 into triangles, as shown in Fig. 128, and then the centre of gravity of each triangle must be found. As soon as this is done, you have only to find the resultant of the forces acting upon the centres of gravity of the triangles; and this will give the centre of gravity of the whole. [If it is required to find the centre of gravity of a triangular pyramid, it will be necessary to draw lines from the angle, a and b, Fig. 129, towards the centres of gravity, e and e', of the opposite triangles. Where these two lines cut each other at e" is the centre of gravity. a Fig. 128. b Fig. 129. Fig. 130. [The centre of gravity of a cone, Fig. 130, with a circular base, lies on the straight line which passes from the apex, a, to the centre of the base, b, and its distance from the central point, b, of the base is of the whole line. [The centre of gravity of a straight line is exactly in the middle.] 144 When the centre of gravity is above the point of suspension, there is produced a pressure upon that point. When the centre of gravity is beneath the point of suspension, there is produced a pull upon that point. The stability of bodies is intimately connected with the position of their centre of gravity. A body may be in a condition, 1st, of indifferent; 2nd, of stable; 3rd, of instable equilibrium. Indifferent equilibrium ensues when a body is supported upon its centre of gravity; for then it is immaterial what position is given to it—it remains in all at rest. Stable equilibrium ensues when the point of support is above the centre of gravity. If the body be disturbed from this situation, it oscillates for a time, and finally returns to its original position. Instable equilibrium is exhibited when the point of support is beneath the centre of gravity. The body being movable, in this instance it revolves upon its point of support, and turns into such a position that its centre of gravity comes immediately beneath that point. In the theory of the balance, hereafter to be described, these facts are of the greatest importance. When bodies are supported upon a basis, their stability depends on the position of their line of direction. The line of direction has already been defined to be a line drawn from the centre of gravity perpendicularly downward. If the line of direction falls within the basis of support, the body remains supported. If the line of direction falls outside the basis of support, the body overturns. Thus, let there be a block of wood or metal, Fig. 131, of which c is the centre of gravity, c d the line of direction, and let it be supported on its lower face, a b; so long as the line of direction falls within this basis, the block remains in equilibrio. Again, let there be another block, Fig. 132, of which c is the centre of gravity, and c d the line of direction. Inasmuch as this falls outside of the basis, a b, the body overturns: A ball upon a horizontal plane has its line of direction within its point of support; it therefore rests indifferently in any position in which it may be laid. But a ball upon an inclined plane has its line of direction outside its point of support, and therefore it falls continually. From similar considerations we understand the nature of the difficulty of poising a needle upon its point. The centre of gravity is above the point of support, and it is almost impossible to adjust things so that the line of direction will fall within the basis. The slightest inclination instantly causes it to overturn. When the centre of gravity is very low, or near the basis, there is more difficulty in throwing the line of direction outside the basis than when it is Fig. 131. с Fig. 132. Fig. 133. high. For this reason carriages which are loaded very high, or have much weight on the top, are more easily overturned than those the load of which is low, and the weight arranged beneath, as is shown in Fig. 133. The stability of a body is greater according as its weight is greater, its centre of gravity lower, and its basis wider. The principles here laid down apply to the case of the flotation of bodies. When an irregular-shaped solid mass is placed on the surface of a fluid, it arranges itself in a certain position, to which it will always return if it be purposely overset. In many such solids another position may be found, in which they will float in the liquid; but the slightest touch overturns them. Bodies, therefore, may exhibit either stable or unstable flotation. A long cylinder floating on one end is an instance of the latter case, but if floating with its axis parallel to the surface of the liquid, of the former. These phenomena depend on the relative positions of the centre of gravity of the floating solid, and that of the portion of liquid which it displaces. The G former retains an invariable position as respects the solid mass, but the latter shifts in the liquid as the solid changes its place. Equilibrium takes place when the centre of gravity of the floating body and that of the portion of liquid displaced are in the same line of direction. If of the two the former is undermost, stable equilibrium ensues; but if it is above the centre of gravity of the displaced liquid, unstable equilibrium takes place. To this, however, there is an exception-it arises when a body floats on its largest surface. There are two forces involved in the determination of the position of flotation: 1st, the gravity of the body downward; 2nd, the upward pressure of the liquid. The former is to be referred to the centre of gravity of the body itself, and the latter takes effect on the centre of gravity of the displaced liquid. If these two centres are in the same vertical line, they counteract each other; but in any other position a movement of rotation must ensue. The solid, therefore, turns over, and finally comes into such a position as satisfies the conditions of equilibrium. On these principles a cube will float on any one of its faces, and a sphere in any position whatever; but if the sphere be not of uniform density, one part of it being heavier than the rest, motion takes place until the heaviest part is lowest. A long cylinder floating on its end is unstable, but when it floats lengthwise, stable. It is obvious that these principles are of great importance in ship-building, and the loading and ballasting of ships. CHAPTER XXV. THE PENDULUM, Simple and Physical Pendulums-Nature of Oscillatory Motion-Centre of Oscillation-Laws of Pendulums Cycloidal Vibrations-The Seconds' Pendulum-Measures of Time, Space, and Gravity-Compensation Pendulums. A SOLID body suspended upon a point with its centre of gravity below, so that it can oscillate under the influence of gravity, is called a pendulum. A simple pendulum is imagined to consist of an imponderable line, having freedom of motion at one end, and at the other a point possessing weight. A physical pendulum consists of a heavy metallic ball, suspended by a thread or slender wire. The position of rest of a pendulum is when its centre of gravity is perpendicularly beneath its point of suspension; its length, therefore, is in the line of direction. If it be removed from this position, it returns to it again after making several oscillations backward and forward. Its descending motions are due to the gravitating action of the earth, its ascending due to its own inertia. A pendulum once in motion would vibrate continually were it not for friction on its point of suspension, the rigidity of the thread, if it be supported by one, and the resistance of atmospheric air. The length of a pendulum is the distance that intervenes between its Q с b Fig. 134. point of suspension and its centre of nous an oscillation is the magnitude of the arc, a d, expressed in degrees, minutes, and seconds. The time of an oscillation is the time necessary for the pendulum to traverse this arc.] Let a b c, Fig. 135, be a pendulous body, supported on the ca point, a, and performing its oscillations upon that point. If we Fig. 135. consider the motions of two points, such as b and c, it will appear that under the influence of gravity the point, b, which is nearer to the point of suspension, would perform its oscillations more quickly than the point, c. But inasmuch as in the pendulous body both are supposed to be inflexibly connected together, by reason of the solidity of the mass, both are compelled to perform their oscillations in the same time. The point, b, will, therefore, tend to accelerate the motions of c, and c will tend to retard the motions of b. It follows, therefore, that in every pendulum there is a point the velocity of which, multiplied by the mass of the pendulum, is equal to the quantity of motion in the pendulum. To this point the name of centre of oscillation is given. In a linear pendulum-that is, a rod of inappreciable thickness— the centre of oscillation is two-thirds the length from the point of suspension. In a right-angled conical mass the centre of oscillation is at the centre of the base. The centre of oscillation possesses the remarkable property that it is convertible with the centre of suspension-that is to say, if a pendulum vibrates in a given time, when supported on its ordinary centre of suspension, it will vibrate in the same time exactly if it be suspended on its centre of oscillation. Advantage has been taken of this property to determine the lengths of pendulums with great precision, and thereby the intensity of gravity and the figure of the earth. In these cases a simple bar of metal, of proper length, with knife-edges equidistant fromi ts ends, has been used, and adjust *This word is derived from the Greek isos, equal, and chronos, time. The term signifies equality of time. ment made until the bar vibrated equally when supported on either knifeedge. The distance between the knife-edges is the length of the pendulum. Pendulums of equal lengths vibrate in the same place in equal times, provided their angles of elongation do not exceed two or three degrees. Pendulums of unequal lengths vibrate in unequal times the shorter more quickly than the longer-the times being to one another as the square roots of the lengths of the pendulums. B If we take a circle, B, Fig. 136, and, causing it to roll along a plane, B D, mark out the path which is described by a point, P, in its circumference, the line so marked is designated a cycloid.* P Fig. 136. When a pendulum vibrates in a cycloid, it will describe all arcs thereof in equal times; and the time of each oscillation is to the time in which a heavy body would fall through half the length of the pendulum as the circumference of a circle is to its diameter. The difference, therefore, between oscillation in cycloidal and circular arcs is, that in the former all oscillations are isochronous, but in the latter they are not; for the larger the circular are the longer, the time of oscillation. And as circular movement is the only one which can be conveniently resorted to in practice, it is necessary to reduce circular to cycloidal oscillations by calculation. When the length of the pendulum is such that its time of oscillation is equal to one second, it is called a seconds' pendulum. This length differs at different places. [The following table shows the length of the seconds' pendulum, in English inches, at several places where it has been tried. It is taken from Mr. Airey's treatise on the "Figure of the Earth," in the "Encyclop. Metrop." Some allowance must be made for the error in the correction for the density of the air, about 0018 for each.] Under the equator it is shorter than at the poles; and this evidently arises from the circumstance that the intensity of gravity, as has been already explained, is different at those points; for the figure of the earth not being a perfect sphere, but an oblate spheroid, its polar axis being shorter than its equatorial, a body at the poles is more powerfully attracted than one at the *This word is derived from the Greek kuklos, a circle, and eidos, like. The term signifies a curve generated by the rotation of a circle along a right line. |