7. If the weight of the empty vessel in Fig. 10, (1), is small compared with the weight of the contained water, will the force required to lift the vessel and water be greater or less than the force exerted by the water against the bottom? Explain. 8. A whale when struck with a harpoon will often dive straight down as much as 400 fathoms (2400 ft.). If the body has an area of 1000 sq. ft., what is the total force to which it is subjected? 9. A hole 5 cm. square is made in a ship's bottom 7 m. below the water line. What force in kilograms is required to hold a board above the hole? 10. Thirty years ago standpipes were generally straight cylinders. To-day they are more commonly of the form shown in Fig. 9. What are the advantages of each form? PASCAL'S LAW 22. Transmission of pressure by liquids. From the fact that pressure within a free liquid. depends simply upon the depth and density of the liquid, it is possible to deduce a very surprising conclusion, which was first stated by the famous French scientist, mathematician, and philosopher, Pascal (1623-1662). (1) b Let us imagine a vessel of the shape shown in Fig. 10, (1), to be filled with water up to the level ab. For simplicity let the upper portion be assumed to be 1 square centimeter in cross section. Since the density of water is 1, the force with which it presses against any square centimeter of the interior surface which is h centimeters beneath the level ab is h grams. Now let 1 gram of water (that is, 1 cubic centimeter) be poured into the tube. Since each square centimeter of surface, which before was h centimeters beneath the level of the FIG. 10. Proof of Pascal's law a (2) water in the tube, is now h +1 centimeters beneath this level, the new pressure which the water exerts against it is h+1 grams; that is, applying 1 gram of force to the square centimeter of surface ab has added 1 gram to the force exerted by the liquid against each square centimeter of the interior of the vessel. Obviously it can make no difference whether the pressure which was applied to the surface ab was due to a weight of water or to a piston carrying a load, as in Fig. 10, (2), or to any other cause whatever. We thus arrive at Pascal's conclusion that pressure applied anywhere to a body of confined liquid is transmitted undiminished to every portion of the surface of the containing vessel. 1000 kg 1 kg 23. Multiplication of force by the transmission of pressure by liquids. Pascal himself pointed out that with the aid of the principle stated above we ought to be able to transform a very small force into one of unlimited magnitude. Thus, if the area of the cylinder ab (Fig. 11) ab is 1 sq. cm., while that of the cylinder AB is 1000 sq. cm., a force of 1 kg. applied to ab would be transmitted by the liquid so as to act with a force of 1 kg. on each square centimeter of the surface AB. Hence the total upward force exerted against the piston AB by the 1 kg. applied at ab would be 1000 kg. Pascal's own words are as follows: A vessel full of water is a new principle in mechanics, and a new machine for the multiplication of force to any required extent, since one man will by this means be able to move any given weight." ec FIG. 11. Multiplication of force by transmission of pressure 24. The hydraulic press. The experimental proof of the correctness of the conclusions of the preceding paragraph is furnished by the hydraulic press, an instrument now in common use for subjecting to enormous pressures paper, cotton, etc. and for punching holes through iron plates, testing the strength of iron beams, extracting oil from seeds, making dies, embossing metal, etc. Hydraulic presses of great power have been designed for use in steel works to replace huge steam hammers. Compressing forces of 10,000 tons or more are thus obtained. Much cold steel, as well as hot, is now pressed instead of hammered. Such a press is represented in section in Fig. 12. As the small piston p is raised, water from the cistern C enters the piston chamber through the valve v. As soon as the downstroke begins, the valve v closes, the valve v opens, and the pressure applied on the piston p is transmitted through the tube K to the large reservoir, where it acts on the large cylinder P. The force exerted upon P is as many times that applied to p as the area of P is times the area of p. K FIG. 12. Diagram of a hydraulic press 25. No gain in the product of force times distance. It should be noticed that, while the force acting on AB (Fig. 11) is 1000 times as great as the force acting on ab, the distance through which the piston AB is pushed up in a given time is but 1000 of the distance through which the piston ab moves down. For forcing ab down a distance of 1 centimeter crowds but 1 cubic centimeter of water over into the large cylinder, and this additional cubic centimeter can raise the level of the water there but 100 centimeter. We see, therefore, that the product of the force acting by the distance moved is precisely the same at both ends of the machine. This important conclusion will be found in our future study to apply to all machines. 1 26. The hydraulic elevator. Another very common application of the principle of transformation of pressure by liquids is found in the hydraulic elevator. The simplest form of such an elevator is shown in Fig. 13. The cage A is borne on the top of a long piston P which runs in a cylindrical pit C of the same depth as the height to which the carriage must ascend. Water enters the pit either directly from the water mains, m, of the city's supply or, if this does not furnish sufficient pressure, from a special reservoir on top of the building. When the elevator boy pulls up on the cord cc, the valve opens so as to make connection from m into C. The elevator then ascends. When cc is pulled down, v turns so as to permit the water in C to escape into the sewer. The elevator then descends. Where speed is required the motion of the piston is communicated indirectly to the cage by a system of pulleys like that shown in Fig. 14. m POSITION OF U POSITION OF V A FIG. 13 FIG. 14 Diagrams of hydraulic elevators With this arrangement a foot of upward motion of the piston P causes the counterpoise D of the cage to descend 2 feet, for it is clear from the figure that when the piston goes up 1 foot, enough rope must be pulled over the fixed pulley p to lengthen each of the two strands a and b 1 foot. Similarly, when the counterpoise descends 2 feet, the cage ascends 4 feet. Hence the cage moves four times as fast and four times as far as the piston. The elevators in the Eiffel Tower in Paris are of this sort. They have a total travel of 420 feet and are capable of lifting 50 people 400 feet per minute. The cylinder C and piston P are often not in a pit but lie in a horizontal position. Most modern elevators are electric rather than hydraulic. 27. City water supply. Fig. 15 illustrates the method by which a city is often supplied with water from a distant source. The aqueduct from the lake a passes under a road r, a brook b, and a hill H, and into a reservoir e, from which it is forced by the pump p into the standpipe P, whence it is distributed to the houses of the city. If a static condition prevailed in f B FIG. 15. City water supply from lake the whole system, then the water level in e would of necessity be the same as that in a, and the level in the pipes of the building B would be the same as that in the standpipe P. But when the water is flowing, the friction of the mains causes the level in e to be somewhat less than that in a, and that in B less than that in P. It is on account of the friction both of the air and of the pipes that the fountain ƒ does not rise nearly as high as the ideal limit shown in the figure. QUESTIONS AND PROBLEMS 1. A jug full of water may often be burst by striking a blow on the cork. If the surface of the jug is 200 sq. in. and the cross section of the cork 1 sq. in., what total force acts on the interior of the jug when a 10-lb. blow is struck on the cork? 2. How does your city get its water? How is the pressure in the pipes maintained? |