THE PRINCIPLE OF ARCHIMEDES * 28. Buoyant effect of a liquid. The preceding experiments have shown that an upward force acts against the bottom of any body immersed in a liquid. If the body is a boat, a cork, a piece of wood, or any body which floats, it is clear that since it is in equilibrium this upward force must be equal to the weight of the body. Even if the body does not float, everyday observation shows that it still loses a portion of its natural weight; for it is well known that it is easier to lift a stone under water than in air, or, again, that a man in a bathtub can support his whole weight by pressing lightly against the bottom with his fingers. It was, indeed, this very observation which first led the old Greek philosopher Archimedes (287-212 B.C.) to the discovery of the exact law which governs the buoyant effect of a liquid upon a body placed in it. Hiero, the tyrant of Syracuse, had ordered a gold crown made, but suspected that the artisan had fraudulently used silver as well as gold in its construction. He ordered Archimedes to discover whether or not this was true. How to do so without destroying the crown was at first a puzzle to the old philosopher. While in his daily bath, noticing the loss of weight of his own body, it suddenly occurred to him that any body immersed in a liquid apparently loses a weight equal to the weight of the displaced liquid. He is said to have jumped at once to his feet and rushed through the streets of Syracuse crying "Eureka! Eureka!" (I have found it! I have found it!). 29. Theoretical proof of Archimedes' principle. It is probable that Archimedes, with that faculty which is so common among men of great genius, saw the truth of his conclusion without going through any logical process of proof. Such a proof, however, can easily be given. Thus, the upward force F on the bottom of the block B (Fig. 20) is equal to the * A laboratory exercise on the experimental proof of Archimedes' principle should either precede or accompany this discussion. See, for example, Experiment 5 of "Exercises in Laboratory Physics," by Millikan, Gale, and Davis. weight of the column of liquid C. The downward force F' on the top of this block is equal to the weight of the column of liquid C'. It is, then, clear that the upward force must exceed the downward force by the weight of the liquid whose volume is equal to that of the block. Archimedes' principle may be stated thus: The buoyant force exerted by a liquid is exactly equal to the weight of the displaced liquid. FIG. 20. Proof that an immersed body is buoyed up by a force equal to the weight of the displaced liquid n m The reasoning is exactly the same, no matter what may be the nature of the liquid in which the body is immersed, nor how far the body may be beneath the surface. Further, if the body weighs more than the liquid which it displaces, it must sink; for it is urged down with the force of its own weight, and up with the lesser force of the weight of the displaced liquid. But if it weighs less than the displaced liquid, then the upward force due to the displaced liquid is greater than its own weight, and consequently it must rise to the surface. When it reaches the surface, the downward force upon the top of the block, due to the liquid, becomes zero. The body must, however, continue to rise until the upward force on its bottom is equal to its own weight. But this upward force is always equal to the weight of the displaced liquid; that is, to the weight of the column of liquid mben (Fig. 21). Hence FIG. 21. Proof that a floating body is buoyed up by a force equal to the weight of the displaced liquid A floating body displaces its own weight of the liquid in which it floats. This statement is embraced in the statement of Archimedes' principle, for a body which floats has lost its whole weight (see opposite page). This huge iceberg was formed by the movement into the sea of the great glacial ice cap of Greenland. The submarine, one of the newest of marine inventions, is a simple application of the principle of Archimedes, one of the oldest principles of physics. In order to submerge, the submarine allows water to enter her ballast tanks until the total weight of the boat and contents becomes nearly as great as that of the water she is able to displace. The boat is then almost submerged. When she is under headway in this condition, a proper use of the horizontal, or diving, rudders sends her beneath the surface, or, if submerged, brings her to the surface, so that she can scan the horizon with her periscope. The whole operation takes but a few seconds. When the submarine wishes to come to the surface for recharging her batteries or for other purposes, she blows compressed air into her ballast tanks, thus driving the water out of them. Submarines are propelled on the surface by Diesel oil engines; underneath the surface, by storage batteries and electric motors To test our reasoning for this case, place an overflow can (Fig. 22) on a trip scale, fill it with water, and carefully balance it. Now float a block of wood in the can. When the overflow of water ceases, the scales again balance. What do you conclude? 30. Specific gravity of a heavy solid. The specific gravity of a body is by definition the ratio of its weight to the weight of an equal flow can volume of water (§ 16). Since a submerged FIG. 22. An overbody displaces a volume of water equal to its own volume, however irregular it may be, Specific gravity of body: = weight of body weight of water displaced Making application of Archimedes' principle, we have weight of body buoyancy (or loss of weight in water) 31. Specific gravity of a solid lighter than water. If the body is too light to sink of itself, we may still obtain the weight of the equal volume of water by forcing it be- W2 FIG. 23. Method of finding specific gravity of a light solid body alone (or its loss of weight in water) and is therefore equal to the weight of the displaced water. |