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heat may be supposed given to 91 pounds of water, and will therefore just suffice to raise the temperature of the ball and the water 2 degrees of Fahrenheit's thermometer.

626. The principle of the Conservation of Energy is not confined to the two subjects which we have considered and here brought into connexion, namely mechanical work and heat; it is extended by philosophers so as to include chemical action and electrical action: the principle asserts that in all transformations of Energy from one kind of action to another the amount of Energy remains unchanged.

627. The earlier researches into the subject of Energy related chiefly to the conversion of Work into Heat; more recently attention has been given to the conversion of Heat into Work. Sir W. Thomson has been led to a principle which is called the Dissipation of Energy, meaning however something different from what the words would at first naturally suggest. It is found that although we can easily convert Work into Heat, we cannot get all the Heat back again into the form of Work. In consequence of this it is held that the mechanical Energy of the universe is becoming every day more and more changed into Heat; and so science looks forward "to an end when the whole universe will be an equally heated inert mass, and from which every thing like life or motion or beauty will have utterly gone away." Two treatises have been published on the subject of Energy to which the student may refer for more information; The Conservation of Energy... by Professor Balfour Stewart, and An Elementary Exposition of the Doctrine of Energy, by D. D. Heath.

LIV. ELASTICITY.

628. In the theory of Mechanics we suppose for simplicity that we are concerned with rigid bodies, that is with bodies which retain always the same shape and size. But a body is never really rigid; it always changes more or less its shape and size under the action of force, and when the force is withdrawn the body resumes, at least to some extent, the original shape and size: the property by virtue of which this resumption takes place is called Elasticity.

629. Gaseous bodies and liquids may be said to be elastic inasmuch as they regain their original size when any pressure to which they have been exposed is withdrawn; but we now propose to confine ourselves to the case of solid bodies, in which the shape as well as the size have to be considered.

630. A solid is said to be perfectly elastic which returns exactly to its original size and shape, when any constraint to which it has been subjected is removed; and it is said to be imperfectly elastic when this is not the case. Strictly speaking no solid is perfectly elastic, though some solids possess the property of elasticity in a very high degree, as for example, Indian rubber, ivory, glass, and marble; other solids, as lead and clay, have very little elasticity. If a ball of ivory be allowed to fall on a slab of polished marble it will rebound to nearly the original height. It is believed that during the brief time of collision the ball was at first slightly flattened, and then resumed its original form; and that the rebound is occasioned by the effort to resume its original form.

631. Practically speaking almost every solid body may be considered perfectly elastic up to a certain point. That is, there is generally a limit of constraint for every body to which it may be exposed and from which it will recover when the constraint is removed, the recovery being complete so far as our means of observation extend. But if the constraint is carried beyond this limit the body undergoes some appreciable lasting change of shape or of size, or of both; in technical language the body receives a permanent set. For degrees of constraint beyond the limit the body is imperfectly elastic. It is obvious that in practice it will be necessary to pay great attention to the limit of elasticity, so as to ensure safety and durability in constructions. The perfect elasticity of some bodies within certain limits is shewn by obvious facts; thus a steel watch-spring, or the spring by which a pen-knife is closed, will continue to work for years without any appreciable change. We proceed to consider three different modes of constraint to which bodies may be exposed, and to state the laws which determine the behaviour of bodies

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

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