181. Other cases of elasticity. All cases of deformation can be stated in terms of the moduli which have been defined. In advanced treatises on elasticity, formulas are worked out for the bending of beams under various conditions, the stability of loaded pillars, the elongation and bending of various types of springs, and many other practical problems. In the case of a beam the bending will depend on the shape of the cross section of the beam and on how the ends are fastened. If the loaded beam is supported at the ends, with the ends free (that is, not clamped), the upper surface of the beam will be compressed and the lower surface stretched. The theory will involve only the elasticity of stretch. When the beam is rectangular in cross section, as in the case of a heavy plank, and a load w is applied at the center, theory gives, as the depression of the center, where I is the distance between the supports, b the breadth, d the vertical thickness, and M Young's modulus for the material used. If two planks are laid one on top of the other, and securely nailed together, so that one cannot slip over the other, the vertical thickness of the resulting beam is twice as great as that of one plank. But the formula shows that the bending varies inversely as the cube of the thickness. Hence the bending would be one eighth of that of a single plank. The beam is much stronger than if it had not been nailed together. For if the planks are not nailed together, we have two beams instead of one, each taking approximately half the load, and the two bending approximately half as much as one plank alone. 182. Ultimate stress or strength. By the term ultimate stress, or ultimate strength, is meant the greatest stress which can be produced in a body without rupturing it. Ultimate tension is the maximum stress that can be applied to stretch a body without rupturing it. It is often called tensile strength. The maximum compressional stress that can be applied to a body without breaking it is usually called compressional strength. It is customary in specifying the values of all these qualities to state them in terms of force per unit area. PROBLEMS 1. A load of 3000 dynes is hung by a wire 1 mm. in radius. Compute the stress. 2. A load of 2000 lb. is supported by a rod half an inch in diameter. Compute the stress. 3. A rod 20 ft. long is stretched one fourth of an inch. Compute the strain. 4. A wire 0.5 mm. in diameter and 3 m. long is stretched 3.0 mm. What is the strain ? 5. How much will a steel wire 1 mm. in radius and 2 m. long be stretched when it supports a mass of 10 kg.? (M=21 × 1011 dynes/cm.2) 6. How much will an iron rod with radius 0.1 in. and length 10 ft. be stretched by a load of 200 lb.? (M = 27 × 106 lb./in.2) 7. Compute Young's modulus from the data given in section 173. 8. A rod 100 in. long and 2 sq. in. in cross section supports a stretching load of 30 tons. If Young's modulus is 30 × 100 lb./in.2, compute the elongation. 9. A vertical rod 2 m. long and 0.2 sq. cm. in cross section is stretched 0.08 cm. The load is the weight of 100 kg. Find (a) the stress, (b) the strain, and (c) Young's modulus. 10. What is the fractional change in the volume of a glass ball produced by a pressure of 3000 lb./in.2? (K= 6×106 lb./in.2) 11. A steel shaft 30 ft. long, diameter 1 in., rotates at 300 R. P. M. One end of the shaft is twisted 0.1 radians. (a) How much is the torque? (n = 12 × 106 lb./in.2) (b) What power is transmitted? 1 PART II. HEAT CHAPTER XVI TEMPERATURE AND ITS MEASUREMENT Temperature, 183. Temperature scales, 184. Thermometers, 185. Maximum and minimum thermometers, 186. The clinical thermometer, 187. Resistance thermometers, 188. Gas thermometers, 189. Thermoelectric thermometers, 190. Measurement of high temperatures, 191. Standard temperatures, 192. 183. Temperature. Our sense organs have taught each of us the meaning of the words cold and hot. The word temperature refers to the same thing as hotness, and is used when we have in mind some method for its determination more accurate than our sense organs. In temperature measurements some physical property of a substance which varies with temperature must be used. In our common thermometers the change in volume of a liquid is used. Some other methods will be described later. 184 Temperature scales. In any temperature scale it is necessary to define at least two fixed points on the scale. The two commonly used are the temperature at which pure ice melts and the temperature at which pure water boils under standard barometric pressure. A hundred years ago there were a large number of different temperature scales in use, but this number has diminished until there are at present only three scales in common use. These are the Fahrenheit, used chiefly in English-speaking countries; the centigrade, used universally in scientific work, and as the domestic scale in some countries of Europe; and the Réaumur, used commonly in certain European countries (for example, Sweden, Germany, and Russia) and in some industries elsewhere. The "fixed points" of these scales are defined as follows: From these values one can see that the interval between the two fixed points is divided in the centigrade scale into 100 divisions, in the Fahrenheit into 180, and in the Réaumur into 80. Hence 100 divisions centigrade = 180 Fahrenheit = 80 Réaumur. These numbers are in the ratio of 5, 9, 4. From this the student should be able to deduce the following rules: F=C+32, C=(F-32), R=(F-32), where F is the temperature on the Fahrenheit scale, C the reading of the same temperature on the centigrade scale, and R that on the Réaumur scale. The absolute, or Kelvin scale, differs from the centigrade only in the position of the zero point, its divisions being the same size. Its zero point is – 273° C. The gas scale is practically identical with the Kelvin scale. The relation between the centigrade and Kelvin scales is very simple: K=C+273. The Kelvin scale is used only in scientific work, especially in dealing with very high or very low temperatures and in many computations involving the laws of gases and the laws of radiation. 185. Thermometers. Practically all substances expand with an increase of temperature and contract when the temperature falls. All common thermometers utilize this property. The mercury or alcohol in a glass thermometer rises when the temperature increases because its volume increases more than the volume of the glass container. If glass expanded more than the liquid, the thermometer would fall with a rise of temperature. The mercury thermometer is more accurate than the spirit thermometer, but its visibility is not so good. The mercury thermometer must have a tube of smaller bore than the spirit thermometer; hence it is usually harder to see. The reason the mercury column must be more narrow is that mercury expands at about one sixth the rate at which alcohol does. In any careful work the thermometer used should be carefully compared with a standard thermometer. Often in the laboratory the student is expected to test the ice point by submerging the thermometer in crushed ice and to test the boiling point by placing it in the steam arising from boiling water. Practically all the larger laboratories of physical science have standard thermometers which have been carefully tested at the Bureau of Standards or some other testing laboratory. F -20 -10 D F -90 -80 -i' 0 C -70 10 -60 20 -50 30 40 40 30 i 50 B 20 186. Maximum and minimum thermometers. There are many occasions when it is important to know the highest or lowest temperature that has been reached during a certain period. For example, one might wish to know the lowest temperature that has been reached during the night. A maximum thermometer is one that indicates the highest, or maximum, temperature to which it has been subjected since it was set; a minimum thermometer is one that indicates the lowest, or minimum, temperature. Fig. 137 shows diagrammatically one form of thermometer which serves as both a maximum and a minimum thermometer. The large bulb B and the tube as far as the point A contain alcohol or some other suitable liquid. The tube from A to C contains mercury. Above C there is more alcohol, which extends up into D, but only partly fills it, thus giving room for expansion. When the temperature rises, FIG. 137 |