ELASTICITY OF BODIES.

29

Putty and wet clay are instances of bodies which are almost entirely destitute of elasticity.

The resistance of a cylindrical or prismatic bar to elongation or flexure is measured by a number called "Young's modulus of elasticity." The following are examples of its value for different substances, in kilogrammes per square millimetre:

To illustrate the meaning of this table by the case of steel; a steel wire whose section is a square millimetre will be elongated by 21793 of its length by a weight of one kilogramme. The elongation is inversely proportional to the section and directly proportional to the stretching weight; so that a steel wire whose section is half a millimetre will, when stretched by a weight of six kilogrammes, receive twelve times the elongation above specified.

The resistance of a cylindrical or prismatic bar or beam to bending (called its flexural rigidity) is proportional to the value of Young's modulus of elasticity for the material of the bar or beam; so that from the dimensions of the bar, the value of this modulus, and the magnitudes and directions of the externally applied forces, the amount of bending could be calculated.

The resistance of a cylindrical rod to twisting (called torsional rigidity) does not depend upon the value of Young's modulus, but upon an entirely distinct element, an element which is sometimes called simply "rigidity," and which expresses the resistance which a square of given thickness would oppose to being changed by external forces into a rhombus of the same area, having angles differing by a given small amount from right angles.

Elasticity being a molecular phenomenon, it is to be expected that all circumstances which modify the molecular constitution of a body will alter its elasticity; but in the present state of science it is impossible to predict à priori the nature and direction of the change, the effects being sometimes opposite for different substances. Thus tempering (that is to say heating followed by sudden cooling), which, as is well known, augments in a high degree the hardness and elasticity of steel, produces a reverse effect on the bronze of which gongs are made. This alloy, in fact, when cooled slowly, possesses the fragility of glass; whilst, when cooled suddenly, it can he wrought with the hammer.

The elasticity of springs furnishes a simple means of comparing forces. Fig. 14 represents an apparatus designed for this purpose and called a dynamometer. It is formed of two plates of steel, AB and A'B', jointed at their extremities to two metallic bridles which connect them. To the middle of the upper plate is attached a ring, by means of which the apparatus can be suspended from a fixed

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point. To the middle of the lower plate is attached a hook, which can either receive a weight or serve as a point of application for the force which is to be tested. Under the action of the force thus applied the spring plates bend, the distance of the middle points increases, and this increase serves to measure the force itself; or the force may be measured by observing what weight must be suspended from the hook to produce the same effect.

The spring-balance is another apparatus of the same kind. In its most common form it contains a spiral spring which is elongated by the application of the force which is to be measured. The equality of the graduations illustrates the law above stated of the proportionality of distortions to the forces producing them. The resistance of a spiral spring to elongation depends chiefly (as shown by Professor James Thomson) on the torsional rigidity of the wire which composes it.

It is important to remark, that whereas a pair of scales is essentially a measure of mass, a spring-balance is essentially a measure of force. Hence if a spring-balance be graduated so as to show weights correctly at a medium latitude, it will indicate too little if carried to the equator (where the force of gravity is feebler), and too much at the poles (where gravity is more intense).

CHAPTER IV.

GRAVITY.

24. Terrestrial gravity is the force in virtue of which all bodies fall to the surface of the earth. This force is general; its effects are observed in all places and for all bodies. If some of these latter, as smoke and hydrogen gas, appear to be exceptions, it is because they are sustained by the air in the same manner as cork is sustained by water. In space deprived of air, not only do all bodies fall, but, as we shall see later, they fall with equal velocities.

25. Direction of Gravity.-The direction of gravity is called the vertical. It is easily determined by the aid of the simple apparatus called a plumb-line, which consists of a thread fixed at one end and carrying a heavy body at the other. When the system is in equilibrium it is clear that the resultant of the actions of gravity on all the parts of the heavy body has exactly the same. direction as the thread, since it is this which prevents the fall. But it can be shown that this direction does not change when the form and volume of the heavy body are altered, it must therefore be the same as the direction of the force which would act upon one of the elementary particles if suspended alone at the extremity of the thread.

It can be shown by experiment that the direction of gravity is perpendicular to the surface of a liquid in equilibrium, or to use the common expression, to the surface of still water. For this purpose a plumb-line OA is suspended over the

surface of a fluid in equilibrium (which should be slightly opaque,

Fig. 16.-Experiment for showing that the Plumb-line is perpendicular to the surface of a fluid at rest.

as blackened water), and the plummet is allowed to plunge in the liquid. The image AB of the thread produced by reflection at the surface of the liquid will be seen with great distinctness, and will be observed to be exactly in a line with the thread itself. Now we shall see in a subsequent part of this treatise, that whenever reflection takes place at a plane mirror, each point of the object and the corresponding point in the image are on the same perpendicular to the mirror and at equal distances from the mirror on opposite sides. Since, then, in the experiment here described the thread and its image are

in one straight line, this line must be perpendicular to the surface.

Fig. 17.-Verticals at different places.

The surface of still water defines in each locality what is called the surface of the earth. This expression denotes the surface of an imaginary ocean of calm water supposed to cover the whole earth. This surface is known to be sensibly spherical. It follows that the different verticals will nearly meet in the centre of the earth. The figure shows the relative position of some verticals CZ, CZ, CZ"; it is evident that they contain angles equal to the an

gular distance which separates the corresponding places.

At any one locality all verticals may be treated as parallel, on account of the immense distance of the centre of the earth. Let us calculate, for example, the angle contained between two verticals a metre (39.37 inches) apart. Ten millions of metres correspond to a quarter of the earth's circumference, that is to say, to 90°. A length of a metre, therefore, represents 90° divided by ten millions, that is to say, about 13 of a second, a quantity quite inappreciable even with our most perfect instruments. It should be remarked, however, that the parallelism of the verticals at any one place is a physical fact, completely independent of all previous knowledge of the figure of the earth, and can be established by direct observation.

We may remark in passing, that the latitudes of places on the earth's surface are determined by the directions of the verticals. What is commonly called the latitude of a place is the angle which a vertical at the place makes with a plane perpendicular to the earth's axis of rotation. As distinguished from this, the geocentric latitude (which is required in a few astronomical problems) is the angle which a line drawn from the place to the earth's centre makes with a plane perpendicular to the axis of rotation. The difference between common and geocentric latitude generally amounts to some minutes, and attains its greatest value (11′ 29′′) at latitude 45°. 26. Point of Application of Gravity-Centre of Gravity.-Gravity being a property of matter, its points of application must evidently be the different material particles which compose each body. Though a body be divided into as many parts as we please, and even reduced to the state of impalpable powder, each of the grains thus obtained will be subject to the action of gravity. The total force which urges a body to fall is the sum, or more strictly, the resultant of all the forces which are thus

actually applied to its several elements.

P

Gravity on the different Points of a
Body.

Now these forces are parallel, as has just Fig. 18.-Parallelism of the Forces of been stated, and act in the same direction; their resultant is therefore equal to their sum, and it constitutes what is called the weight of the body; that is to say, the force with which it presses the obstacle which prevents it from falling. The point of application G of this resultant