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3. If the water pressure in the city mains is 70 lb. to the square inch, how high above the town is the top of the water in the standpipe ?
4. The cross-sectional areas of the pistons of a hydraulic press were 3 sq. in. and 60 sq. in. How great a weight would the large piston sustain if 75 lb. were applied to the small one?
5. The diameters of the pistons of a hydraulic press were 2 in, and 20 in. What force would be produced upon the large piston by 50 lb. on the small one?
6. The water pressure in the city mains is 80 lb. to the square inch. The diameter of the piston of a hydraulic elevator of the type shown in Fig. 13 is 10 in. If friction could be disregarded, how heavy a load could the elevator lift? If 30% of the ideal value must be allowed for frictional loss, what load will the elevator lift?
7. Suppose a tube 5 mm. square and 200 cm. long is inserted into the top of a box 20 cm. on a side and filled with water; what will be the total force on the bottom of the box? on the top?
THE PRINCIPLE OF ARCHIMEDES *
28. Apparent loss of weight of a body in 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, cork, 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.) (see opposite page 22) to the discovery of the exact law which governs the loss of weight of a body in a liquid.
* A laboratory exercise on the experimental proof of Archimedes' principle should either precede or accompany this discussion. See, for example, Experiment 6 of the authors' Manual.
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 were 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 must apparently lose 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, since the upward force on the bottom of the block abcd (Fig. 16) is equal to the weight of the column of liquid obce, and since the downward force on the top of this block is equal to the weight of the column of liquid oade, it is clear that the upward force must exceed the down- is buoyed up by a ward force by the weight of the column of liquid abcd. Archimedes' principle may be stated thus:
FIG. 16. Proof that an immersed body
force equal to the weight of the displaced liquid
The buoyant force exerted by a liquid is exactly equal to the weight of the displaced liquid.
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 on the top of the block, due to the liquid, becomes zero. The body must, however, continue
FIG. 17. Proof that
a floating body is buoyed up by a force equal to the weight of the displaced liquid
to rise until the upward
ways equal to the weight of the displaced
A floating body must displace 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.
FIG. 18. Method of weighing a body under water
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 volume of water (§ 17). Since a submerged body displaces a volume of water equal to its own volume, however irregular it may be,
Specific gravity of body = Weight of water displaced Weight of body Making application of Archimedes' principle, we have
Specific gravity of body
Weight of body
Loss of weight in water
Fig. 18 shows a common method of weighing under water.
The celebrated geometrician of antiquity; lived at Syracuse, Sicily; first made a determination of T and computed the area of the circle; discovered the laws of the lever and was author of the famous saying, "Give me where I may stand and I will move the world"; discovered the laws of flotation; invented various devices for repelling the attacks of the Romans in the siege of Syracuse; on the capture of the city, while in the act of drawing geometrical figures in a dish of sand, he was killed by a Roman soldier to whom he cried out, "Don't spoil my circle"
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