Page:The American Cyclopædia (1879) Volume VI.djvu/497

ELASTICITY 489 will extend the bars to double their original length, if we suppose the material to withstand such a force without breaking ; the correspond- ing numbers are found in the third column of the preceding table, and constitute for each substance the so-called " Young's modulus of elasticity," which may be defined as the weigbt that will so elongate a bar of unit square sec- tion as to double its original length. The moduli given in the third column in kilo- grammes per square millimetre may be con- verted into the British system of pounds per square inch by multiplying by 1,422 '3, as given in the fourth column. It is frequently con- venient to express the above given weight modulus by an equivalent length of a bar of the same material ; this number, which may be called the length modulus, is found by dividing the weight modulus by the weight of the unit of length, or in the French system, by the specific gravity of the substance and multiply- ing the quotient by 100,000. Thus a bar of glass will be stretched to double its length by the weight of a similar bar of glass whose length is 19,000 metres. In general, for large forces and changes of form that leave a perma- nent deformation of the solid, the resistance to crushing differs from the resistance to stretch- ing; this is shown by the coefficients of strength for these two kinds of forces ; and the differ- ence of bodies in this respect is a highly im- portant element in calculating their fitness for building or mechanical purposes. On the other hand, the effect of an exceedingly small force in producing a slight compression is sen- sibly equal to the effect of the same force in producing a slight extension; therefore the above given moduli of elasticity hold good for such values of both pressure and tension as do not approach the limit of perfect elasticity. If now a bar of any substance be slightly bent, its convex side is extended, but its concave side compressed ; there must therefore be with- in its substance some neutral line, or axis of no change whatever ; this line constitutes the elastic curve of Bernoulli, and has been the subject of many investigations. The exact curvature of this line, or of the bar, is known when the applied forces are given. The for- mulas relating to the various cases that occur in engineering are given in works on- applied mechanics, such as those of Weisbach and Rankine. One of the most interesting and important applications of the elasticity of a plane curved spring is found in the case of the mainspring and especially of the hair spring of a chronometer; the theory of their action has been studied, among others, by Yvon Villarceau (1863). Other applications are found in the dynamometer for measuring great tensions, the horizontal fixed glass thread for weighing minute quantities, the tuning fork and the stretched strings of musical instruments, &c. The near approach to perfect elasticity that is the property of some bodies, as steel and glass, is evinced by the perfect uniformity of the times of vibration of the strings, bells, bars, &c., that give out pure musical notes on being struck, and that no matter how extensive their arcs of vibration or how loud their cor- responding notes ; if the elastic force did not increase precisely in proportion to the amount of the molecular displacement, we should notice discords instead of the single pitch that ema- nates from the sounding body. The perfect uniformity of these minute vibrations has sug- gested their application in the sphygmograph and other chronographs, where they replace the pendulum for the measurement of minute intervals of time; indeed, in the chronograph of M. Hipp, a vibrating spring serves also as the regulator of the revolving cylinder. The vibrations of a carefully arranged metallic spring, hermetically sealed in an exhausted glass vessel, and at a standard temperature, depending as they do solely on the elasticity of the body, offer a means of measuring small intervals of time with almost absolute perfec- tion ; and Sir William Thomson has suggested that for scientific purposes such an arrangement must be adopted as a standard far more con- stant than the rotation of the earth or the vi- bration of a pendulum. If a straight bar be twisted about its longitudinal axis, an elastic force is brought into play to resist the me- chanical couple by which the twisting is effect- ed. This, the elasticity of torsion, does not depend upon that of compression or extension ; its modulus must therefore be determined by direct experiment. The laws of the elasticity of torsion have been studied originally by Coulomb, and subsequently by Binet, J. Thom- son, and especially by St. Venant (1855) and Thomson and Tait. The principal applications of torsional elasticity are found in the spring balance in ordinary commercial use, and in the torsion balance used in electrical measure- ments, in the Cavendish experiment, and other delicate researches. In the latter instrument the amount of torsion is evidently a direct measure of the external force applied and to be measured. In the former instrument (the spring balance) the elastic wire is coiled around a cylinder like the threads of a screw, and an external force is applied parallel to the axis of the cylinder to increase or diminish the dis- tance between the coils. The theory of the action, of such spiral or helicoidal springs has been developed by Binet (1814), and by Prof. J. Thomson (1848), by whom it has been shown that the force that opposes the elongation or contraction of a helicoidal spring is the elas- ticity of torsion. Besides the applications above mentioned of the spring balance to commercial uses, it has been proposed to use a very delicate instrument of this construction as a means of investigating the local and general variations of gravity on the earth's surface, for which purpose it possesses some advantages over the pendulum. The grandest application of the laws of the elasticity of solids that has ^ yet been made consists in the investigation into