By the end of this section, you will be able to:
- Define pressure in terms of weight
- Explain the variation of pressure with depth in a fluid
- Calculate density given pressure and altitude
If your ears have ever popped on a plane flight or ached during a deep dive in a swimming pool, you have experienced the effect of depth on pressure in a fluid. At the Earth's surface, the air pressure exerted on you is a result of the weight of air above you. This pressure is reduced as you climb up in altitude and the weight of air above you decreases. Under water, the pressure exerted on you increases with increasing depth. In this case, the pressure being exerted upon you is a result of both the weight of water above you and that of the atmosphere above you. You may notice an air pressure change on an elevator ride that transports you many stories, but you need only dive a meter or so below the surface of a pool to feel a pressure increase. The difference is that water is much denser than air, about 775 times as dense.
Consider the container in Figure 11.13. Its bottom supports the weight of the fluid in it. Let us calculate the pressure exerted on the bottom by the weight of the fluid. That pressure is the weight of the fluid divided by the area supporting it, which is the area of the bottom of the container.
We can find the mass of the fluid from its volume and density
The volume of the fluid is related to the dimensions of the container. It is
where is the cross-sectional area and is the depth. Combining the last two equations gives
If we enter this into the expression for pressure, we obtain
The area cancels, and rearranging the variables yields
This value is the pressure due to the weight of a fluid. The equation has general validity beyond the special conditions under which it is derived here. Even if the container were not there, the surrounding fluid would still exert this pressure, keeping the fluid static. Thus the equation represents the pressure due to the weight of any fluid of average density at any depth below its surface. For liquids, which are nearly incompressible, this equation holds to great depths. For gases, which are quite compressible, one can apply this equation as long as the density changes are small over the depth considered. Example 11.4 illustrates this situation.
Example 11.3 Calculating the Average Pressure and Force Exerted: What Force Must a Dam Withstand?
In Example 11.1, we calculated the mass of water in a large reservoir. We will now consider the pressure and force acting on the dam retaining water. (See Figure 11.14.) The dam is 500 m wide, and the water is 80 m deep at the dam. (a) What is the average pressure on the dam due to the water? (b) Calculate the force exerted against the dam and compare it with the weight of water in the dam, previously found to be .
Strategy for (a)
The average pressure due to the weight of the water is the pressure at the average depth of 40 m, since pressure increases linearly with depth.
Solution for (a)
The average pressure due to the weight of a fluid is
Entering the density of water from Table 11.1 and taking to be the average depth of 40 m, we obtain
Strategy for (b)
The force exerted on the dam by the water is the average pressure times the area of contact.
Solution for (b)
We have already found the value for . The area of the dam is , so that
Although this force seems large, it is small compared with the weight of the water in the reservoir—in fact, it is only of the weight. Note that the pressure found in part (a) is completely independent of the width and length of the lake—it depends only on its average depth at the dam. Thus the force depends only on the water's average depth and the dimensions of the dam, not on the horizontal extent of the reservoir. In the diagram, the thickness of the dam increases with depth to balance the increasing force due to the increasing pressure.epth to balance the increasing force due to the increasing pressure.
Atmospheric pressure is another example of pressure due to the weight of a fluid, in this case due to the weight of air above a given height. The atmospheric pressure at the Earth's surface varies a little due to the large-scale flow of the atmosphere induced by the Earth's rotation, which creates weather highs and lows). However, the average pressure at sea level is given by the standard atmospheric pressure , measured to be
This relationship means that, on average, at sea level, a column of air above of the Earth's surface has a weight of , equivalent to (See Figure 11.15.)
Example 11.4 Calculating Average Density: How Dense Is the Air?
Calculate the average density of the atmosphere, given that it extends to an altitude of 120 km. Compare this density with that of air listed in Table 11.1.
If we solve for density, we see that
We then take to be atmospheric pressure, is given, and is known, and so we can use this to calculate .
Entering known values into the expression for yields
This result is the average density of air between the Earth's surface and the top of the Earth's atmosphere, which essentially ends at 120 km. The density of air at sea level is given in Table 11.1 as or about 15 times its average value. Because air is so compressible, its density has its highest value near the Earth's surface and declines rapidly with altitude.
Example 11.5 Calculating Depth Below the Surface of Water: What Depth of Water Creates the Same Pressure as the Entire Atmosphere?
Calculate the depth below the surface of water at which the pressure due to the weight of the water equals 1 atm.
We begin by solving the equation for depth .
Then we take to be 1 atm and to be the density of the water that creates the pressure
Entering the known values into the expression for gives
Just 10.3 m of water creates the same pressure as 120 km of air. Since water is nearly incompressible, we can neglect any change in its density over this depth.
What do you suppose is the total pressure at a depth of 10.3 m in a swimming pool? Does the atmospheric pressure on the water's surface affect the pressure below? The answer is yes. This seems only logical, since both the water's weight and the atmosphere's weight must be supported. So the total pressure at a depth of 10.3 m is two atm—half from the water above and half from the air above. We shall see in Pascal's Principle that fluid pressures always add in this way.