part of it we please, provided the part is such as could be cut off by a plane. We have only to keep that part with which we are concerned just below the surface of the water, and observe how much of the water runs over if the vessel were originally full, or through what space the level rises if the vessel were originally only partly filled. 395. There are however practical difficulties which may obstruct the process in the case of some bodies. Thus the solid may be soluble in water; then perhaps some other liquid may be found in which the solid is not soluble. Or the water may make its way into the pores of the substance, as it would within a sponge; then perhaps a thin coat of varnish can be applied sufficiently durable to keep out the wet during the short time occupied by the process. XXXII. WEIGHTS OF SOLIDS IMMERSED 396. In the preceding Chapter we have treated of the immersion of solids in liquids as affording a method of determining the volumes of solids. In that Chapter there is no mechanical principle involved; the whole is a matter of mensuration, that is of elementary Geometry: but we are now about to introduce the reader to some very important mechanical facts. Let us suppose that a person takes a stone weighing about 5 pounds, fastens a string to it, and holds the other end of the string; then he supports the stone, that is he exerts a force sufficient to balance the weight of 5 pounds. Let him now hold the string so that the stone may be immersed in a bucket of water; if the stone rests on the bottom of the bucket it is supported without the exertion of any force by the person. But let us suppose the stone not to touch the bottom of the bucket; in this case the weight is apparently much less than before the stone was immersed, and will seem to the person holding the string to be about 3 pounds. The fact is one which can be easily verified to any extent, and it is universally found that when a heavy body is thus suspended in a liquid in which it would sink if left alone its weight seems diminished; the heavy body is as it were to some degree supported by the liquid. It is customary to say that the solid loses a portion of its weight. 397. The next point to settle is the amount of this diminution of weight. The following is found to be the law: when a solid is suspended in a liquid the weight is diminished by the weight of an equal bulk of the liquid. Or instead of saying by the weight of an equal bulk of the liquid we may say by the weight of the liquid displaced. This law can be easily verified. In the case of the stone which we considered in the preceding Article the weight can be accurately determined before immersion. Again, when the stone is immersed let the end of the string instead of being held by the hand be fastened to the end of the arm of a balance, or to a spring which serves as a weighing machine; thus the apparent weight can be accurately determined. Therefore, by subtraction, the diminution of weight becomes known. And, as in the preceding Chapter, we can find the bulk of the liquid which is equal to the bulk of the solid; and consequently the weight of so much liquid becomes known. From these results we can make the requisite comparison, and thus the truth of the law which we have stated is established. 398. Besides the direct comparison of weights by which, as we have shewn in Art. 397, the truth of the law is established, there are indirect methods by which we obtain the same result. Before the stone is immersed its whole weight is supported by the hand. Suppose the sides of the vessel are vertical. When the stone is immersed the weight of which the hand is relieved must be thrown on the vessel in some way, and we may naturally infer that in consequence there must be an increase of pressure on the base just equal to this, and therefore the same increase in the pressure of the vessel on the ground or on any supports on which it rests. Take a vessel full of water and attach it to some weighing machine; suspend the stone in the vessel; then water runs out equal in bulk to the stone, but the spring weighing-machine does not alter its reading. The relief afforded to the weight of the stone immersed is thus inferred to be exactly equal to the weight of an equal bulk of the water. 399. Or we may establish the truth of the law by reasoning. Suppose a solid immersed in a liquid. The resultant force of the liquid on the solid may be naturally taken to be exactly equal to what it would be on any body of precisely the same size and shape as the solid whatever might be the material of which it was composed. Hence it would be exactly equal to the resultant action on so much of the liquid itself as would occupy just the same space as the solid; and therefore, by Art. 382, this resultant action is a force upwards equal to the weight of the liquid displaced. Thus the diminution of the weight of the suspended solid is equal to the weight of the liquid displaced. 400. We have hitherto supposed the solid entirely immersed in the liquid; but similar considerations apply to the case of a solid partially immersed. The diminution of weight will be equal to the weight of so much liquid as agrees in bulk with the immersed portion of the solid; or we may say briefly that the diminution of weight is equal to the weight of the displaced liquid. 401. Next suppose we have a solid that does not sink but floats on the liquid. In this case the whole weight is lost, that is the whole weight is supported by the liquid. Hence by Art. 400 we see that when a solid floats on a liquid the weight of the solid is equal to the weight of the liquid which it displaces. 402. It will be convenient to state some facts relating to the weight and volume of water which are wanted for numerical applications. The grain is thus determined: a cubic inch of pure water weighs 252-458 grains. is A pound Avoirdupois contains 7000 grains. A cubic foot of water weighs 1728 × 252458 grains, that 7000 that this number to three decimal places is 997-137. Thus it is usually sufficient in practice to take 1000 ounces Avoirdupois as the weight of a cubic foot of pure water. A gallon is a measure which will hold 10 pounds Avoirdupois of pure water, that is 70000 grains. Hence the number of cubic inches in a gallon is 70000 252.458 ; it will be found that this number to three decimal places is 277-274. Thus it is usually sufficient in practice to take 277 as the number of cubic inches in a gallon. XXXIII. APPLICATIONS. 403. The principles of the preceding Chapter lead to various interesting applications and illustrations. One of the most important is the comparison of the weights of equal bulks of various substances. The specific gravity of a body is the proportion which the weight of the body bears to that of an equal bulk of some standard substance; and the standard substance is usually pure water at the temperature of 62 degrees of Fahrenheit's thermometer. The mode of determining the specific gravity of solids is in principle that of Art. 397, with due precaution to ensure accuracy. The solid is weighed, the weight of an equal bulk of the water is found; and the former result divided by the latter gives the specific gravity. We shall recur to the process hereafter, and shall consider also the specific gravity of liquids and gases; we give here a few of the results which have been obtained with respect to solids; the figures are to be found to three decimal places in various works; but with much diversity, so that it will be sufficient here to go to one decimal place. Thus platina is 21.5 times heavier than water, bulk for bulk; gold is 19'4 times heavier; and so on. Some of these results are liable to a little modification under circumstances; thus for hammered gold the specific gravity is nearly 194, and for cast gold it is 19.26. Since a cubic foot of water weighs very nearly 1000 ounces Avoirdupois we can immediately determine the weights of known volumes of other substances, with sufficient practical accuracy, by means of a Table of Specific Gravities. For instance a cubic foot of iron will weigh 78 times as much as a cubic foot of water, that is it will weigh 7.8 times 1000 ounces, that is 7800 ounces. Hence a cubic inch of iron 7800 will weigh 404. Platina and gold are comparatively scarce substances, so that we are limited in the use of very dense materials, that is of materials which are extremely heavy for a given bulk. The power of man over matter is in many ways great; in particular we shall see when we describe the air pump, and some other machines, that we can obtain matter in a state of extreme tenuity; but as yet no means have been found for obtaining matter in a state of extreme density. It is obvious that there are various useful applications which might be made of a very dense substance if such could be readily procured. Thus in forming sea walls, or the foundations of the arches of bridges and other constructions under water, a stone as dense as gold if it could be easily found or composed artificially would be of great service. So, too, a strip of extremely dense material would be very advantageous for the keel of a ship. 405. The support, whether partial or total, which a solid body in a liquid receives from the liquid is a very familiar fact, and is often expressed by the term buoyancy. It is obvious that the degree of support depends on the nature of the liquid. Thus the Table in Art. 403 shews that silver, lead, copper, iron and tin will all sink in water, and all float in mercury. So also as oil is a little lighter than water, bulk for bulk, a body might sink in oil which would float on water. 406. The human body when the chest is expanded by drawing in air is rather lighter than an equal bulk of water, so that it would float on water with some portion not immersed. A person who floats has to exercise care to preserve the mouth and the nose above the surface of the water, so as to secure the power of breathing. When the air is expelled from the chest the bulk of the body is sensibly diminished while the weight suffers no appreciable change; and then the body sinks a little in the water. |