If, instead of being fixed at one end, the two ends of the rod are free, but are supported on two knife-edges placed at a distance L1 apart, and is the distance through which the centre of the rod is deflected when loaded with a force P, Young's modulus is given by

It will be noticed that in this case we are practically dealing with two rods each of length L1/2, fixed as in the first case, and each acted upon by a force P 2 in the upward direction at the point where the rod rests on the knife-edges.

174. Torsional Rigidity.-If one end of a cylindrical wire of radius r and length / is kept fixed while a twisting couple u is applied to the other end, and under this twisting stress the end of the wire turns through an angle o, it is found that so long as is not too great, this angle is proportional to the applied couple u, so that if the couple is doubled, the angle through which the end of the wire is twisted is also doubled. The value of p, in terms of the dimensions of the wire, is given by the equation

where n is a constant depending on the nature of the material of the wire, and is called the simple rigidity or coefficient of torsional rigidity of the wire. It will be noticed, since is inversely proportional to the fourth power of the radius of the wire, that the deflection produced by a given couple increases very rapidly as the radius of the wire decreases. Thus if the radius of the wire is reduced to a half, the value of p, corresponding to the same value of the deflecting couple, increases sixteen-fold. The importance of this rapid decrease of the torsional rigidity of a wire, when the diameter is reduced, comes in when we use the rigidity of such a wire to measure small forces and couples, as in the Cavendish experiment. By very rapidly drawing out a small stick of quartz, raised to a white heat in an oxy-hydrogen blowpipe, Boys has produced threads of fused quartz of such extreme fineness that a force of one dyne acting at each end of a lever I centimetre long (ie. a unit couple) will twist one end of a fibre 10 centimetres long through 360°. The following table gives the coefficient of torsional rigidity for some solids :

175*. Torsion Pendulum.-If a solid body, suspended by a wire, be twisted away from its position of rest and then released, it will execute S.H. vibrations about its position of rest, for the torsional rigidity of the wire will give a force tending to restore the body to its original position proportional to the deflection. If u is the restoring couple due to the rigidity of the wire produced when the body is twisted through unit angle (a radian), and K is the moment of inertia of the solid, then the time of oscillation is given by

or, substituting the value of u in terms of the simple rigidity and dimensions of the wire,

Such a torsional pendulum can be used to prove that Hooke's law holds for torsional strains, that is, that the restoring couple or stress is proportional to the strain or twist, for the time of oscillation is found to be independent of the amplitude of the vibrations, and it is only when the restoring force is proportional to the deflection that this isochrony is secured.

176. Elastic Limit, Elastic Fatigue.-It is found that if a solid is deformed more than a certain amount, then, on the removal of the deforming stress, it does not completely regain its original form. Under these circumstances the body is said to have been strained beyond its elastic limit. The limits within which they may be considered as completely elastic vary very much with different materials. Thus quartz, and to a less extent steel and glass, can suffer a considerable strain, and yet when the stress is removed they will recover their original form; while soft iron, copper, and lead exhibit a permanent deformation or "set" even with quite small strains.

It is found that if the deforming stress is continued for a long time the strain produced gradually increases. This phenomenon is referred to as elastic fatigue, and it seems to show that for long-continued stresses the molecules even of solids gradually take up new configurations. A somewhat similar phenomenon is the fact that after a solid has been strained even below its elastic limit it does not, on the removal of the deforming force, immediately return completely to its original form, but only does so after some time. Thus if a silver wire is twisted in one direction and kept twisted for a day, and then twisted in the opposite direction for an hour, on being released it does not completely recover, but remains slightly twisted in the direction of the last twist. This residual twist gradually disappears, and then a slight twist in the direction of the first one appears, reaches a maximum, and then dies out. The wire thus "remembers" the deformation previously applied, and the residual effects appear in the opposite order to the original deformations.

BOOK II

HEAT

CHAPTER I

THERMOMETRY AND EXPANSION BY HEAT

177. Temperature.-Although we are able in many cases to distinguish by our sensations between hot and cold bodies-for instance, we can by touch often determine which of two bodies is the hotter-yet our senses do not permit of our forming a quantitative estimate of the amount by which one is hotter than the other. In ordinary language we use the words hot, warm, tepid, cool, cold, &c., to indicate a series of states of a body with reference to heat. In scientific language we use the word temperature to express the same series of condition. Thus a hot body is said to have a higher temperature than a cold body.

As we shall see in the following pages, the characteristic which above all others distinguishes bodies of which the temperatures differ is, that if these bodies are placed in contact, then heat will of itself pass from the one to the other until they reach the same temperature. That body which loses heat during the process of equalisation is said to be at the higher temperature.

It is found that not only does the sensation we experience when we touch a body vary with the temperature, but also that most of the physical properties of matter change when the temperature changes. Thus the density, elasticity, refractive index, &c., of a body all depend on the temperature.

In order to have a means of measuring temperature, we make use of the change in some physical property of some kind of matter which takes place as the temperature of the body changes. The physical property which is most often employed for this purpose is the length of a solid, or the volume of a liquid or gas, both of which depend on temperature. In order to define certain fixed temperatures, we also make use of the fact that the physical state of a body depends on the temperature. Thus according to the temperature we may have the same kind of matter existing as a solid, a liquid, or a gas, as, for instance, ice, water, and

steam.

It is found that during the time the change from one state to the other is going on the temperature remains constant. Thus if a quantity of pounded ice is heated over a flame, the whole being kept well stirred, although the ice becomes gradually converted into water the temperature does not rise till the last particle of ice has been melted, the heat supplied by the flame being simply used up in changing the body from the solid state into the liquid state. If, after the ice is all melted, the heating is continued, the water will eventually begin to boil, becoming converted into the gaseous state (steam), and during the change the temperature of the remaining water will remain constant. It will thus be seen that we may use the temperatures at which a given substance, under given conditions, changes its state as fixed points on a scale of temperature.

In order to subdivide the interval between these two temperatures, use is made of the change in volume of some fluid, usually mercury or hydrogen, which occurs with change in temperature. Now there is no

à priori reason for supposing that the rate of change of volume of a substance, say mercury, with temperature is the same at all temperatures. Since, however, we have no special means of measuring temperature as distinct from the effects of temperature on the physical properties of bodies, we have, at any rate as a starting-point, to assume that the rate of change of some fixed property of some standard substance is constant, and to use this change to subdivide the temperature between our two fixed points. For the present, at any rate, we shall take the change in volume of mercury when contained in a glass vessel as the means of defining the temperature between our fixed points.

178. Thermometric Scales.-The lower fixed point of most scales of temperature is the temperature of melting ice under ordinary atmospheric pressure. The upper fixed point is the temperature of the steam given off from water boiling under the pressure of one standard atmosphere (§ 133).

Letv be the apparent increase in volume of a given mass of mercury enclosed in a glass envelope when its temperature is raised from that of melting ice to that of water boiling under standard conditions. Then the interval of temperature which will cause this quantity of mercury to expand by an amount 7/100 is called a degree Centigrade, and is indicated by the symbol, 1° C. On the Centigrade scale (first used by Celsius) the temperature of melting ice is called zero (0° C.), and that of boiling water 100 C., the interval, as has been said, being divided into a hundred degrees. For temperatures below that of melting ice the scale is continued downwards, the sign minus being prefixed. Thus a temperature such that the volume of the above mass of mercury is less than at o°C. is indicated by -5° C. In a similar way the scale is continued above 100° C. In all scientific work, and, with one or two

57 100

exceptions, for everyday use also, the Centigrade scale of temperature is employed in all countries. There are, however, two other scales of temperature occasionally employed. In one of these, the Fahrenheit scale, the temperature of melting ice is called 32° F., and that of water boiling under standard conditions 212° F., the interval being divided into 180°.

The Fahrenheit scale is in use in England for commercial and meteorological purposes. The other temperature scale is one due to Réaumur, on which the temperature of melting ice is called o° R., and that of boiling water 80° R., the interval being divided into 80°.

Any reading on one of these scales can easily be converted to the corresponding reading on either of the others, for from the definitions 100 degrees on the Centigrade scale are equal to 180 degrees on the Fahrenheit scale, and to 80 degrees on the Réaumur scale. Let c, f, and r indicate the readings corresponding to the same temperature on the three scales. Then since o° C. = 32° F.=0° R., if we deduct 32° from the Fahrenheit reading we have the number of degrees on each scale by which the given temperature differs from the temperature of melting ice. Hence

or

c:f-32::: 100: 180: 80,

с f-32 r

100 180 80

By means of these equations, any temperature on any one scale can be immediately converted into either of the other scales.

179. The Mercury Thermometer.-The most commonly employed instrument for the measurement of temperature is the mercury thermometer. The ordinary form of a mercury thermometer is shown in Fig. 143. It consists of a glass bulb B containing mercury, connected B C

A

FIG. 143.

(From Watson's "Fiementary Practical Physics.")

to a stem CA, which is traversed by a fine, uniform capillary bore. Such a thermometer is filled by attaching a small funnel to the open end of the tube, and placing some pure and dry mercury in it. The bulb is then slightly heated, some of the contained air being thus expelled, and, on cooling, a little mercury is driven into the bulb by the atmospheric pressure. This mercury is then heated till it boils, the mercury vapour driving out the air, so that when the bulb cools the mercury is driven in and completely fills the bulb and tube. The thermometer is then heated up to the highest temperature which it is intended to measure, and the end of the tube closed in the blowpipe flame. On cooling, the mercury contracts and the column sinks in the stem, leaving a vacuum