the upper. No appreciable error is made by calculating the pressure at a point in the liquid, on the assumption that the density is uniform throughout. Let the density of the liquid be for a vertical column Now p M = L'; P M = L2 horizontal surface by L height. pg F = L2 horizontal surface by L height, pg F per L horizontal surface = L height. But the pressure is the same in magnitude, whatever be the inclination of the plane of application taken at the point, provided it is sufficiently small; hence pg F per L2 surface = L height. Thus for every unit of height of a column of liquid we have pg times the corresponding unit of pressure. This explains why a pressure is usually expressed as a height; for example, as so many inches of mercury. The equivalence in this case is •491163 pound weight per sq. in. = in. height of mercury at 32° F. EXAMPLES. Ex. 1. Find the relation between pound per sq. inch and kilogramme per sq. centimetre. cm. lb. per sq. inch. Ex. 2. Find the value of the standard atmospheric pressure in terms of dynes per sq. cm. The density of mercury is 13.596 gm, cc., and the standard intensity of gravity is 981 dyne per gm. = Hence 13.596 × 981 dyne per sq. cm. = cm. height of column, the standard height is 76 cm., 76 × 13.596 × 981 dyne per sq. cm. Answer-1·014 x 106 dyne per sq. cm. Ex. 3. At the bottom of a mine a mercurial barometer stands at 77.4 cm.; what would be the height of an oil barometer at the same place, the specific gravity of mercury being 13-596, and that of oil 0.9 ? 13.596 gm. per sq. cm. = cm. of mercury, Ex. 4. The area of the plunger of a force pump being 3 square inches, find the pressure upon it when water is forced up a height of 20 feet. Ex. 5. Find the whole pressure on a vertical lock-gate 14 feet broad, against which the water rises 9 feet. The intensity of the pressure is given by 62.5 lb. weight per sq. ft. ft. height; hence at the bottom it is and at the top, = 9.5 × 62.5 lb. weight per sq. ft., 0 lb. weight per sq. ft. As the pressure increases uniformly with the depth, the average pressure is Now i.e., 9.5 × 62.5lb. weight per sq. ft. 2 14 x 9.5 sq. ft. of surface, 7 × (9.5)2 × 62.5 lb. weight, 39484 lb. weight. Ex. 6. A square board, whose edge is 2 feet long, is immersed in water with one edge just at the surface, and it is inclined to the horizontal at 30°. Find the pressure on one side of it. 62.5 lb weight = ft. long by ft. broad by ft. deep, 1. Reduce pound per square foot and pound per square inch to dyne per square cm., when the intensity of gravity is 981 cm. per sec. per sec. 2. Given the average weight of a man as 12 stones, and the density of a dense crowd as 5 men 7 square feet. Find the pressure per square foot due to a dense crowd on a bridge. = 3. Wind, having a velocity of 50 miles an hour, exerts a pressure of 12 lbs. per square foot. Find the total force exerted on a surface 5 ft. by 6 ft. inclined at an angle of 25° to the direction of the wind. 4. Reduce 25'6, 26°7, and 27 8 inches of mercury to millimetres of mercury. 5. Express a pressure of 100 mm. of mercury in terms of inch of water. 6. Reduce a pressure of 600 millimetres of mercury at 0°C. to dynes per square cm. when the intensity of gravity is 981 cm. per sec. per sec. 7. Express a pressure of 760 mm. of mercury in terms of kilogramme per square metre. 8. Given the density of water as 62.5 lb. per cubic foot, and that 1'025 lb. of sea-water is equivalent in volume to one lb. of pure water. Find the pressure at a depth of 200 feet under the surface of the ocean due to the superincumbent water. 9. If the plunger of a force-pump has a cross-section of 8 square inches, and works 50 feet below the cistern, what pressure is required to force it down? 10. During a storm the barometer at sea-level stood as low as 27 466 inches. What was the pressure in lbs. per square inch? 11. What ought to be the length of a water barometer, inclined to the horizon at an angle of 30°, the mercury barometer standing at 30.5 inches? 12. The diameter of the tube of the barometer is 1 cm., and that of the cistern 4.5 cm. If the mercury in the tube rise through 2.5 cm., find the real alteration in the height of the barometer. 13. What is the theoretical height to which water can be raised by the common pump, when the mercurial barometer stands at 28 inches? 14. A barometer is observed to fall one tenth of an inch when carried up SS feet of vertical height; how much would it fall if taken 132 yards up a hill rising 1 in 3 ? 15. A piece of metal of sp. gr. 8, and weighing 20 lbs. is dropped into a cylinder filled with water; find the additional pressure on the base. 16. What depth of water is required to float an iceberg one mile square by 500 feet high? 17. The neck of a wine bottle with flat bottom is 4 inches long, the total height of the bottle being 12 inches. When the bottle is filled with wine of specific gravity 0.99 to within half an inch of its mouth, what is the pressure on each square inch of the bottom? 18. What is the pressure on a sluice-gate 12 feet broad, against which the water rises 5 feet? 19. A sluice-gate is 4 feet broad and 6 feet deep, and the water rises to a height of 5 feet on one side and 2 feet on the other side. Find the pressure in pounds on the gate. 20. Find the whole pressure upon a vertical dam of a column of water 10 feet deep and 30 feet wide. What would be the pressure of the same head of water against a dam inclined at an angle of 45° to the horizon? 21. A vessel, consisting of a decimetre cube, is filled to one third of its height with mercury, while the rest is filled with water; determine the whole pressure against one of the sides in kilogrammes. 22. A rectangular board, one foot square, is immersed in water with its upper edge 10 feet below the surface of the water, and horizontal, the surface of the board being vertical. Find the total pressure on one side. 23. If the height of the water barometer be 1,033 centimetres, what will be the pressure on a circular disc whose radius is 7 cm. when sunk in water to a depth of 50 metres ? 24. A square plate whose area is 64 square inches is immersed in sea water, its upper edge being horizontal and 12 inches below the surface. Determine the whole pressure of the water on the plate when it is inclined at 45° to the horizon, assuming a cubic inch of sea water to weigh 0'63 ounces. SECTION XXXII.-PRESSURE OF A GAS. ART. 154. Height of Homogeneous Atmosphere. In the case of a vertical column of gas, the density is not uniform throughout. The gas in a horizontal layer is compressed by the weight of the superincumbent gas. It is sometimes convenient to consider what would be the height of a vertical column of gas having the density throughout which it actually has at the bottom, and producing the same pressure at the bottom. The height of such a column is called height of homogeneous atmosphere, because the conception applies to the air of the atmosphere. Prof. Everett suggests the shorter and more appropriate name, "pressure height."* The pressure of the atmosphere is used as a convenient unit of pressure in the same way as the weight of a pound is used as a convenient unit of force. The exact unit is defined by the following equivalences 1 atmosphere 29-922 inches of merc. at 32° F. (British); = = 760 mm. of merc. at 0° C. (French). ART. 155.-Dependence of Density on Pressure. The law discovered by Boyle states that the density of a portion of gas is * Units and Physical Constants, p. 37. |