# Theory of Heat (Maxwell, 4th edition)/Chapter 5

CHAPTER V.
ON THE MEASUREMENT OF PRESSURE AND OTHER INTERNAL FORCES, AND OF THE EFFECTS WHICH THEY PRODUCE.
Every force acts between two bodies or parts of bodies. If we are considering a particular body or system of bodies, then those forces which act between bodies belonging to this system and bodies not belonging to the system are called External Forces, and those which act between the different parts of the system itself are called Internal Forces.
If we now suppose the system to be divided in imagination into two parts, we may consider the forces external to one of the parts to be, first, those which act between that part and bodies external to the system, and, second, those which act between the two parts of the system. The combined effect of these forces is known by the actual motion or rest of the part to which they are applied, so that, if we know the resultant of the external forces on each part, we can find that of the internal forces acting between the two parts.
Thus, if we consider a pillar supporting a statue, and imagine the pillar divided into two parts by a horizontal plane at any distance from the ground, the internal force between the two parts of the pillar may be found by considering the weight of the statue and that part of the pillar which is above the plane. The lower part of the pillar presses on the upper part with a force which exactly counterbalances this weight. This force is called a Pressure. In the same way we may find the internal force acting through any horizontal section of a rope which supports a ​heavy body to be a Tension equal to the weight of the heavy body and of the part of the rope below the imaginary section.
The internal force in the pillar is called Longitudinal Pressure, and that in the rope is called Longitudinal Tension. If this pressure or tension is uniform over the whole horizontal section, the amount of it per square inch can be found by dividing the whole force by the number of square inches in the section.
The internal forces in a body are called Stresses, and longitudinal pressure and tension are examples of particular kinds of stress. It is shown in treatises on Elasticity that the most general kind of stress at any point of a body may be represented by three longitudinal pressures or tensions in directions at right angles to each other.
For instance, a brick in a wall may support a vertical pressure depending on the height of the wall above it, and also a horizontal pressure in the direction of the length of the wall, depending on the thrust of an arch abutting against the wall, while in the direction perpendicular to the face of the wall the pressure is that of the atmosphere.
In solid bodies, such as a brick, these three pressures may be all independent, their magnitude being limited only by the strength of the solid, which will break down if the force applied to it exceeds a certain amount.
In fluids, the pressures in all directions must be equal, because the very slightest difference between the pressures in the three directions is sufficient to set the fluid in motion.
The subject of fluid pressure is so important to what follows that I think it worth while, at the risk of repeating what the student ought to know, to state what we mean by a fluid, and to show from the definition that the pressures in all directions are equal.
Definition of a Fluid.—A fluid is a body the contiguous parts of which act on one another with a pressure which is perpendicular to the interface which separates those parts.
​Since the pressure is entirely perpendicular to the surface, there can be no friction between the parts of a fluid in contact.
P Q be the pressures perpendicular to the sides, and R that perpendicular to the base. Then, since these three forces are in equilibrium, and since R makes equal angles with P and Q, P and Q must be equal. But the forces on which P and Q act are also equal; therefore the pressures referred to unit of area on these faces are equal, which was to be proved.
A great many substances may be found which perfectly fulfil this definition of a fluid when they are at rest, and they are therefore called fluids. But no existing fluid fulfils the definition when it is in motion. In a fluid in motion the pressures at a point may be greater in one direction than in another, or, what is the same thing, the force between two parts may not be perpendicular to the interface which separates those parts.
If a fluid could be found which fulfilled the definition when in motion as well as when at rest, it would be called a Perfect Fluid. All actual fluids are imperfect, and exhibit the phenomenon of internal friction or viscosity, by which their motion after being stirred about in a vessel is gradually stopped, and the energy of the motion is converted into heat.
The degree of viscosity varies from that of tar to that of water, or ether, or hydrogen gas, but no actual fluid is perfect in the sense of the definition when in motion.
​The pressure at any point of a fluid is the ratio of the whole pressure on a small surface to the area of that surface when the area of the surface is made to diminish indefinitely, but so that the centre of gravity of the surface always coincides with the given point.
This pressure is sometimes called hydrostatic pressure, to distinguish it from longitudinal pressure. Both kinds of pressure are measured by the number of units of force in the pressure on unit of area; for instance, in pounds' weight on the square inch or square foot, and in kilogrammes' weight on the square mètre. Both these measures are gravitation measures, and must be multiplied by the value of the intensity of gravity to reduce them to absolute measures.
Pressures are also measured in terms of the height of a column of water or of mercury, which would produce by its weight an equal pressure. Thus a pressure of 16 feet of water is nearly equal to 1,000 pounds' weight on the square foot, and a pressure of 4 inches of water is more nearly equal to 101 grains' weight on the square inch.
In the metrical system the pressure of water on a surface at any depth is expressed by the product of the depth into the area of the surface. If we employ the mètre as the measure of length, the pressure will be expressed in tonnes' weight, but if we use the decimètre, centimètre, or millimètre, the pressure will be expressed in kilogrammes, grammes, or milligrammes respectively, in gravitation measure.
The density of mercury at 0° C. is 13.596 times that of water at 4° C. Hence the pressure due to a given depth of mercury is about 13.6 times that of an equal depth of water.
The Barometer.—The pressure of the air is generally measured by means of the mercurial barometer. This barometer consists of a glass tube closed at one end and filled with mercury, from which all air and moisture are expelled by boiling it in the tube. The tube is then placed with its open end in a vessel of mercury, and its closed end raised till the tube is vertical. The mercury is found to stand at ​a certain level in the tube, the height of which above the level of the mercury in the vessel or cistern is called the height of the barometer.
The surface of the mercury in the cistern is exposed to the pressure of the air, while the surface of the mercury in the tube is exposed only to the pressure of whatever is in the tube above it. The only known substance which can be there is the vapour of mercury, the pressure of which at ordinary temperatures is so small that it may be neglected, so that the pressure of the air may be measured by that due to the difference of level of the mercury in the tube and in the cistern.
The pressure of the atmosphere is, as we know, very variable, and is different in different places; but for various purposes it is convenient to use, as a large unit of pressure, a pressure not very different from the average atmospheric pressure at the mean level of the sea. This unit of pressure is called an atmosphere, and is used in measuring pressures in steam-engines and boilers. Its exact value in the metrical system is the pressure due to a depth of 760 millimètres of mercury at 0° C. at Paris, where the force of gravity is 9.80868 mètres. This is equal to 1.033 kilogrammes' weight on the square centimètre. In absolute measure it is equal to 1,013,237, the gramme, the centimètre, and the second being the fundamental units.
In the British system an atmosphere is defined as the pressure due to a depth of 29.905 inches of mercury at 32° F. at London, where the force of gravity is 32.1889 feet, and is, roughly, 14⁠3/4⁠ pounds' weight on the square inch. It is therefore 0.99968 of the atmosphere of the metrical system.
ON THE ALTERATION OF THE DIMENSIONS AND VOLUME OF BODIES BY MECHANICAL FORCES AND BY HEAT.
We have seen that effects of the same kind in changing the form or volume of bodies are produced by mechanical force and by heat. We cannot therefore fully understand ​the effects of heat alone on these bodies without at the same time considering those of mechanical force.
We have first to explain, from a purely geometrical point of view, the various kinds of change of form of which a body is capable, considering only those cases in which every part of the body undergoes a similar change of form. We shall use the word strain to express generally any alteration of form of a body.
Longitudinal Strain.—Suppose the body to be elongated or compressed in one direction only, so that if two points in the body lie in a line parallel to this direction, their distance will be increased or diminished in a certain ratio, but if the line joining the points be perpendicular to this direction the length of the line will not be altered.
This is called longitudinal extension or compression, or more generally longitudinal strain, and is measured by the fraction of its original length by which any longitudinal line in the body is elongated or contracted.
General Strain.—Such an alteration of the form of the body may take place simultaneously or successively in three directions at right angles to each other. This system of three longitudinal strains is shown in treatises on the motion of continuous bodies to be the most general kind of strain of which a body is capable.
We shall, however, only consider two cases in particular.
1st. Isotropic Strain.—When the strains in the three directions at right angles to each other are all equal, the form of the body remains similar to itself, and it expands or contracts equally in all directions, as most solid bodies do when heated.
Since each of the three longitudinal strains of which this strain is compounded increases the volume by a fraction of itself equal to the value of the longitudinal strain, it follows that when each of the strains is a very small fraction, the total increment of volume is equal to the original volume multiplied by the algebraical sum of the three strains. ​The ratio of the increment of volume to the original volume is called the voluminal expansion when positive, or the voluminal contraction when negative, and it appears, from what we have said, that when the strains are small the voluminal expansion is equal to the sum of the longitudinal extensions, or, when these are equal, to three times the longitudinal extension.
2nd. Shearing Strain.—The other particular case is when the dimensions of the body are extended in one direction in the ratio of $a$ to 1, and contracted in a perpendicular direction in the ratio of 1 to $a$. In this case there is no alteration of volume, but the body is distorted.
WORK DONE BY A STRESS ON A BODY WHOSE FORM IS CHANGING OR IS UNDERGOING A STRAIN.
We shall in the first place suppose that the stress continues constant during the change of form which we consider. If during a considerable change of form the stress undergoes considerable change, we may divide the whole operation into parts, during each of which we may regard the stress as constant, and find the total work by summation.
The general rule is that, if the stress and the strain are of the same type, the work done on unit of volume during any strain is the product of the strain into the average value of the stress.
If, however, the stress be of a type conjugate to the strain, no work is done.
Thus, if the stress be a longitudinal one, we must multiply the average value of the stress by the longitudinal strain in the same direction, and the result is not affected by the magnitude of the longitudinal strains in directions at right. angles to the stress.
If the stress be a hydrostatic pressure, we must multiply the average value of this pressure by the voluminal compression to find the work done on the body per unit of volume, and the result is not affected by any strain of distortion which does not change the volume of the body.
​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/117 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/118 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/119 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/120 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/121 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/122 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/123 ​may be found by drawing P E a tangent to the curve at P, and P F a horizontal line. The portion F E of the vertical line O $p$ cut off between these lines represents, on the scale of pressures, the elasticity of the fluid.
We have hitherto supposed that the temperature of the body remains the same during its compression from the volume P F to the volume Q G. This is the most common supposition when the elasticity of a fluid is to be measured. But in most bodies a compression produces a rise of temperature, and if the heat is not allowed to escape, the effect of this will be to make the increment of pressure greater than in the case of constant temperature. Hence every substance has two elasticities, one corresponding to constant temperature, and the other corresponding to the case where no heat is allowed to escape. The first value is applicable to stresses and strains which are long continued, so that the substance acquires the temperature of surrounding bodies. The second value is applicable to the case of rapidly changing forces, as in the case of the vibrations of bodies which produce sounds, in which there is not time for the temperature to be equalised by conduction. The elasticity in these cases is always greater than in the case of uniform temperature.