under the influence of such constraint and their own elasticity. 632. Extension. If forces are exerted on rods and wires tending to lengthen them the elasticity of the substances will be called into action. Experiments are conducted by fixing one end of a wire to a firm support; then the constraint may be exerted at the other end along the direction of the wire by means of a lever: or the wire may be put in a vertical position and weights attached to the free end. The amount of lengthening thus produced is carefully observed; and the following laws are found to hold so long as the limit of elasticity is not exceeded. (1) Rods and wires are perfectly elastic, that is they resume their original lengths as soon as the stretching force is removed. (2) For the same substance and the same diameter the lengthening is proportional to the original length and also to the stretching force. (3) For rods and wires of the same substance under the same stretching force the lengthening is inversely proportional to the square of the diameter of the rod or wire. 633. The second of the preceding three laws is sometimes called Hooke's Law, from the name of the person who first obtained it; the law does not hold quite strictly however, as we shall see by some numerical results given in the next Chapter. 634. Both calculation and experiment shew that when bodies are lengthened by a stretching force their volumes increase. Thus if a wire is pulled out, and so lengthened, the area of a section of the wire will at the same time diminish, but not so much as to leave the volume just what it was before. It appears in general that all causes which increase the density of a body increase the elasticity, and those which diminish the density diminish the elasticity. Thus the elasticity of metals diminishes continuously as the temperature rises from 59 degrees to 392 degrees of Fahrenheit's thermometer; but iron and steel form exceptions, for their elasticity increases as the temperature rises to 212 degrees, and then diminishes. 635. Compression. In like manner experiments are made on bars or rods by subjecting them to the action of force in the direction of their length which tend to shorten them. Laws similar to those of Art. 632 now hold with respect to the shortening and the compressing force. 636. Torsion. Experiments on the elasticity called into action when wires are twisted are conducted by means of what is called the Torsion Balance. One end of a wire is fixed; the wire hangs vertically, and to the other end a needle is attached at right angles to the wire. Immediately below the needle there is a graduated horizontal circle having its centre in the same vertical line as the wire. By turning the needle round in the horizontal plane, through any angle, the wire is twisted; the angle through which the needle is turned is called the angle of torsion, and the force necessary to retain the needle in the position to which it has been turned is called the force of torsion. When the needle is left to itself after having been turned through any angle it oscillates for some time, to and fro, like a pendulum, until at last it comes to rest in its original position. The elasticity of torsion for stout rods has also been investigated, but by a method different from that used for wires. Both for wires and rods the following laws are found to hold so long as the limit of elasticity is not exceeded. (1) The oscillations for the same rod or wire are, like those of a pendulum, performed in nearly the same time, whether the angle of torsion be greater or smaller. (2) For the same rod or wire the angle of torsion is proportional to the force of torsion. (3) With the same force of torsion, and with rods or wires of the same diameter and of the same substance, the angle of torsion is proportional to the length of the rod or wire. (4) If the same force of torsion is applied to wires of the same length and the same substance the angle of torsion is inversely proportional to the fourth power of the diameter, that is to the square of the square of the diameter. 637. A solid when cut into a rod or a thin plate, and fixed at one end, after having been more or less bent strives to return to its original position. This kind of elasticity is of frequent application in the arts, as for instance in carriage-springs, watch-springs, and springs for measuring weights. The elasticity of hair, wool, and feathers is of service in pillows and cushions. 638. The importance of the elasticity of bodies, especially of the metals, for the ordinary concerns of life is forcibly stated in the Illustrations of Mechanics by the late Professor Moseley. "With the elasticity of metallic bodies every one is conversant. It is a property which, as it belongs to steel, iron, and brass, contributes eminently to the resources of art, and ministers largely to the uses of society. Were it, indeed, not for this property, it would be in vain that the metals should be dug out of the earth and elaborated into various utensils. Infinitely more brittle than glass, they would immediately be dashed to pieces by the slight shocks to which every thing is more or less subject; a shower of hail, or even of rain, would be sufficient to indent their surfaces, and the impact of the minute particles of dust blown against them by the wind would be sufficient permanently to destroy their polish." LV. STRENGTH OF MATERIALS. 639. In all questions of practical engineering it is of the utmost importance to ascertain how far we can rely on the materials we employ to support the strains or pressures which may be brought to bear on them. To a great extent the necessary information consists in numerical results connected with the principles of the preceding Chapter on Elasticity. 640. Modulus of Elasticity. Suppose a given rod or bar held fast at one end and stretched by a force at the other; then by Hooke's Law the amount of lengthening is proportional to the stretching force: see Art. 633. This Law indeed holds only so long as the amount of lengthening is slight; but let us assume for the moment that the Law holds for any amount of lengthening. Then by the T. P. 18 application of a proper force, the rod or bar could just be doubled in length; this force expressed in pounds Avoirdupois per square inch is called the Modulus of Elasticity. The term was introduced by Dr Thomas Young. 641. The values of the Modulus of Elasticity have been determined by experiment for almost every solid substance of importance, and will be found in works on Practical Mechanics, such as Rankine's Applied Mechanics. We give here a selection from these values. 642. The meaning of the foregoing Table will be seen from an example. Suppose a rod or bar of cast brass, one square inch in section; then if a weight of one pound were hung at the end the bar would be lengthened 1 9170000 part of its original length; by two pounds it would be lengthened double this amount; by three pounds triple this amount; and so on. This holds so long as the lengthening is not very great; if it held for any amount of lengthening the length would be just doubled by a weight of 9170000 pounds. If the area of the section of the brass rod is more or less than a square inch, more or less weight will be required in proportion, to produce the same amount of lengthening; thus if it be half a square inch in section half the weight will be required; if it be three square inches in section three times the weight will be required; and so on, 643. Tenacity. Suppose we take a rod or bar of any material; and stretch it by a weight. As the weight is increased so also the amount of lengthening increases, but at last, when the weight becomes sufficiently great, the rod or bar breaks. The breaking weight is taken as a measure of the tenacity of the bar or rod; it is determined by experiment and expressed in pounds Avoirdupois per square inch. 644. The following Table gives the Tenacity of various materials. 645. The following Table also gives in a convenient shape information respecting the tenacity of various metals in the form of a wire. Suppose wires one-sixteenth of an inch in diameter formed of different metals; then the numbers of pounds which they would support are determined by experiment to be approximately the following: Iron 512 Silver 175 646. When iron is stretched beyond the elastic limit the character of the phenomena will depend altogether on the nature of the iron. If the iron is soft and ductile it will be reduced to a much smaller size in the neighbourhood of the point where the fracture ultimately takes place; the area of a section may thus be reduced to threefourths of the original area. This peculiarity is sometimes called toughness; it is in many respects of great value, |