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35. Pressure of Liquids against Surfaces.—The sight of a great ship, perhaps built of iron and floating on water, causes one to wonder at the force that supports it. This same force is noticed when one pushes a light body, as a cork, under water. It is quite evident in such a case that a force exists sufficient to overcome the weight of the cork so that it tends to rise to the surface. Even the weight of our bodies is so far supported by water that many persons can float.

Fig. 17.—Water forces the card against the chimney.

The following experiment provides a means of testing this force:

If an empty can is pushed down into water, we feel at once the force of the liquid acting against the object and tending to push it upward. It may be noticed also that so long as the can is not completely submerged the deeper the can is pushed into the water the greater is the upward force exerted by the liquid.

We may test this action in various ways: a simple way is to take a cylindrical lamp chimney, press a card against its lower end and place it in the water in a vertical position. The force of the water will hold the card firmly against the end of the chimney. (See Fig. 17.) The amount of force may be tested by dropping shot into the tube until the card drops off. At greater depths more shot will be required, showing that the force of the water increases with the depth. Or one may pour water into the chimney. It will then be found that the card does not drop until the level of the water inside the chimney is the same as on the outside. That is, before the card will fall off, the water must stand as high within the chimney as without no matter to what depth the lower end of the chimney is thrust below the surface of the water.

36. Law of Liquid Pressure.—As there is twice as much water or shot in the chimney when it is filled to a depth of 10 cm. as there is when it is filled to a depth of 5 cm. the force of the water upward on the bottom must be twice as great at a depth of 10 cm. as at a depth of 5 cm. Since this reasoning will hold good for a comparison of forces at any two depths, we have the law: "The pressure exerted by a liquid is directly proportional to the depth."

The amount of this force may be computed as follows: First, the card stays on the end of the tube until the weight of water from above equals the force of the water from below, and second, the card remains until the water is at the same height inside the tube as it is outside. Now if we find the weight of water at a given depth in the tube, we can determine the force of the water from below. If for instance the chimney has an area of cross-section of 12 sq. cm. and is filled with water to a depth of 10 cm., the volume of the water contained will be 120 ccm. This volume of water will weigh 120 g. This represents then, not only the weight of the water in the tube, but also the force of the water against the bottom. In a similar way one may measure the force of water against any horizontal surface.

37. Force and Pressure.—We should now distinguish between force and pressure. Pressure refers to the force acting against unit area, while force refers to the action against the whole surface. Thus for example, the atmospheric pressure is often given as 15 pounds to the square inch or as one kilogram to the square centimeter. On the other hand, the air may exert a force of more than 300 pounds upon each side of the hand of a man; or a large ship may be supported by the force of thousands of tons exerted by water against the bottom of the ship.

In the illustration, given in Art. 36, the upward force of the water against the end of the tube at a depth of 10 cm. is computed as 120 grams. The pressure at the same depth will be 10 grams per sq. cm. What will be the pressure at a depth of 20 cm.? at a depth of 50 cm.? of 100 cm.? Compare these answers with the law of liquid pressure in Art. 36.

38. Density.—If other liquids, as alcohol, mercury, etc., were in the jar, the chimney would need filling to the same level outside, with the same liquid, before the card would fall off. This brings in a factor that was not considered before, that of the mass[B] of a cubic centimeter of the liquid. This is called the density of the liquid. Alcohol has a density of 0.8 g. per cubic centimeter, mercury of 13.6 g. per cubic centimeter, while water has a density of 1 g. per cubic centimeter.

39. Liquid Force against Any Surface.—To find the force exerted by a liquid against a surface we must take into consideration the area of the surface, and the height and the density of the liquid above the surface. The following law, and the formula representing it, which concisely expresses the principle by which the force exerted by a liquid against any surface may be computed, should be memorized:

The force which a liquid exerts against any surface, equals the area of the surface, times its average depth below the surface of the liquid, times the weight of unit volume of the liquid.

Or, expressed by a formula, F = Ahd. In this formula, "F" stands for the force which a liquid exerts against any surface, "A" the area of the surface, "h," for the average depth (or height) of the liquid pressing on the surface, and "d", for the weight of unit volume of the liquid. This is the first illustration in this text, of the use of a formula to represent a law. Observe how accurately and concisely the law is expressed by the formula. When the formula is employed, however, we should keep in mind the law expressed by it.

We must remember that a liquid presses not only downward and upward but sideways as well, as we see when water spurts out of a hole in the side of a vessel. Experiments have shown that at a point the pressure in a fluid is the same in all directions, hence the rule given above may be applied to the pressure of a liquid against the side of a tank, or boat, or other object, provided we are accurate in determining the average depth of the liquid; The following example illustrates the use of the law.

For Example: If the English system is used, the area of the surface should be expressed in square feet, the depth in feet and the weight of the liquid in pounds per cubic foot. One cubic foot of water weighs 62.4 lbs.

Suppose that a box 3 ft. square and 4 ft. deep is full of water. What force will be exerted by the water against the bottom and a side?

From the law given above, the force of a liquid against a surface equals the product of the area of the surface, the depth of the liquid and its weight per unit volume, or using the formula, F = Ahd. To compute the downward force against the bottom we have the area, 9, depth, 4, and the weight 62.4 lbs. per cubic foot. 9 × 4 × 62.4 lbs. = 2246.4 lbs. To compute the force against a side, the area is 12, the average depth of water on the side is 2, the weight 62.4, 12 × 2 × 62.4 lbs. = 1497.6 lbs.