Specific gravity is the same in all systems, since it simply expresses how many times as heavy as an equal volume of water a body is. Density, however, which we have defined as the mass per unit volume, is different in different systems. Thus, in the English system the density of iron is 462 pounds per cubic foot (7.4 x 62.4), since we have found that water weighs 62.4 pounds per cubic foot and that iron weighs 7.4 times as much as an equal volume of water.* QUESTIONS AND PROBLEMS † 1. A liter of milk weighs 1032 grams. What is its density and its specific gravity? 2. A ball of yarn was squeezed into of its original bulk. What effect did this produce upon its mass, its volume, and its density? 3. If a wooden beam is 30 × 20 × 500 cm. and has a mass of 150 kg., what is the density of wood? 4. Would you attempt to carry home a block of gold the size of a peck measure? (Consider a peck equal to 8 1. See table, p. 8.) 5. What is the mass of a liter of alcohol? 6. How many cubic centimeters in a block of brass weighing 34 g.? 7. What is the weight in metric tons of a cube of lead 2 m. on an edge? (A metric ton is 1000 kilos, or about 2200 lb.) 8. Find the volume in liters of a block of platinum weighing 45.5 kilos. 9. One kilogram of alcohol is poured into a cylindrical vessel and fills it to a depth of 8 cm. Find the cross section of the cylinder. 10. Find the length of a lead rod 1 cm. in diameter and weighing 1 kg. * Laboratory exercises on length, mass, and density measurements should accompany or follow this chapter. See, for example, Experiments 1, 2, and 3 of the authors' Manual. † Questions and problems to supplement this chapter and all following chapters are given in the Appendix, page 447. CHAPTER II PRESSURE IN LIQUIDS LIQUID PRESSURE BENEATH A FREE SURFACE 18. Force beneath the surface of a liquid. We are all conscious of the fact that in order to lift a kilogram of mass we must exert an upward pull. Experience has taught us that the greater the mass, the greater the force which we must exert. The force is commonly taken as numerically equal to the mass lifted. This is called the weight measure of a force. A push or pull which is equal to that required to support a gram of mass is called a gram of force. Thus, five grams of force are needed to lift a new five-cent piece. To investigate the nature of the forces beneath the free surface of a liquid we shall use a pressure gauge of the form shown in Fig. 4. If the rubber diaphragm which is stretched across the mouth of a thistle tube A is pressed in lightly with the finger, the drop of ink B will be observed to move forward in the tube T, but it will return again to its first position as soon as the finger is removed. If the pressure of the finger is increased, the drop will move forward a greater distance than before. We may therefore take the amount of motion of the drop as a measure of the force acting on the diaphragm. Now let A be pushed down first 2 cm., then 4 cm., then 8 cm. below the surface of the water (Fig. 4). The motion of the index B will show that the upward force continually increases as the depth increases. Careful measurements made in the laboratory will show that the force is directly proportional to the depth.* *It is recommended that quantitative laboratory work on the law of depths and on the use of manometers accompany this discussion. See, for example, Experiments 4 and 5 of the authors' Manual. Let the diaphragm A (Fig. 4) be pushed down to some convenient depth (for example, 10 centimeters) and the position of the index noted. Then let it be turned sidewise so that its plane is vertical (see a, Fig. 4), and adjusted in position until its center is exactly 10 centimeters beneath the surface, hat is, until the average depth of the diaphragm is the same as before. The position of the index will show that the force is also exactly the same as before. FIG. 4. Gauge for measuring liquid pressure Let the diaphragm then be turned to the position b, so that the gauge measures the downward force at a depth of 10 centimeters. The index will show that this force is again the same. We conclude, therefore, that at a given depth a liquid presses up and down and sidewise on a given surface with exactly the same force. 19. Magnitude of the force. If a vessel like that shown in Fig. 5 is filled with a liquid, the force against the bottom is obviously equal to the weight of the column of liquid resting upon the bottom. Thus, if F represents this force in grams, A the area in square centimeters, h the depth in centimeters, and d the density in grams per cubic centimeter, we shall have F= Ahd. (1) FIG. 5 Since, as was shown by the experiment of the preceding section, the force is the same in all directions at a given depth, we have the following general rule: The force which a liquid exerts against any surface is equal to the area of the surface times its average depth times the density of the liquid. It is important to remember that "average depth" means the vertical distance from the level of the free surface to the center of the area in question. 20. Pressure in liquids. Thus far attention has been confined to the total force exerted by a liquid against the whole of a given surface. It is often more convenient to imagine the surface divided into square centimeters or square inches, and then to consider the force on one of these units of area. In physics the word "pressure" is used exclusively to denote the force per unit area. Pressure is thus a measure of the intensity of the force acting on a surface, and does not depend at all on the area of the surface. Since, by § 19, F= Ahd, and since by definition the pressure p is equal to the force per unit area, we have F A = (2) Therefore the pressure at a depth of h centimeters below the surface of a liquid of density d is hd grams per square centimeter. If the height is given in feet and the density in pounds per cubic foot, then the product hd gives pressure in pounds per square foot. Dividing by 144 gives the result in pounds per square inch. hd. = 21. Levels of liquids in connecting vessels. It is a perfectly familiar fact that when water is poured into a teapot it stands at exactly the same level in the spout as in the body of the teapot; or if it is poured into a number of connected vessels like those shown in Fig. 6, the surfaces of the liquid in the various vessels lie in the same horizontal plane. Now the pressure at c (Fig. 7) was shown by the experiment of § 18 to be equal to the density of the liquid times the depth cg. The pressure at o in the opposite direction must be equal to SIY FIG. 6. Water level in communicating vessels that at c, since the liquid does not tend to move in either direction. Hence the pressure at o must be ks times the density. If water is poured in at s so that the height ks is increased, the pressure to the left at o becomes greater than the pressure to the right at c, and a flow of water takes place to the left until the heights are again equal. LA k FIG. 7. Why water seeks its level It follows from these observations on the level of water in connected vessels that the pressure beneath the surface of a liquid depends simply on the vertical depth beneath the free surface, and not at all on the size or shape of the vessel. QUESTIONS AND PROBLEMS 1. Soundings at sea are made by lowering some kind of pressure gauge. When this gauge reads 1.3 kg. per square centimeter, what is the depth? (Density of sea water=1.026.) 2. Kerosene is 0.8 as heavy as water (1 cu. ft. of water=62.4 lb.). Find the pressure of the kerosene per square foot and per square inch on the bottom of an oil tank filled to a depth of 30 ft. 3. What pressure per square inch is required to force water to the top of the Woolworth building in New York City, 780 ft. high? 4. A swimming tank 50 ft. square is filled with water to a depth of 5 ft. Find the force of the water on the bottom; on one side. 5. If the areas of the surfaces AB in Fig. 8, (1) and (2), are the same, and if water is poured into each vessel at D till it stands at the same height above AB, how will the downward force on AB in Fig. 8, (2), compare with that in Fig. 8, (1)? Test your answer, if possible, by making AB a piece of cardboard and pouring water in at D, in each case, until the cardboard is forced off. 6. If the vessel shown in Fig. 10, (1) (p. 15), has a base of 200 sq. cm. and if the water stands 100 cm. deep, what is the total force on the bottom? |