The earth’s atmosphere is not a laboratory-size container. The height of the atmosphere is such that the gravitational contribution to pressure is important. As FIGURE 14.7 shows, the density of air slowly decreases with increasing height until approaching zero in the vacuum of space. Consequently, the pressure of the air, what we call the atmospheric pressure $p_{\text{atmos}}$, decreases with height. The air pressure is less in Denver than in Miami.

FIGURE 14.7 The pressure and density decrease with increasing height in the atmosphere.

The atmospheric pressure at sea level varies slightly with the weather, but the global average sea-level pressure is 101,300 Pa. Consequently, we define the standard atmosphere as

The standard atmosphere, usually referred to simply as “atmospheres,” is a commonly used unit of pressure. But it is not an SI unit, so you must convert atmospheres to pascals before doing most calculations with pressure.

Given that the pressure of the air at sea level is 101.3 kPa, you might wonder why the weight of the air doesn’t crush your forearm when you rest it on a table. Your forearm has a surface area of $\approx 200{\text{cm}}^2=0.02{\text{m}}^2$, so the force of the air pressing against it is $\approx 2000\text{N}$ $(\approx 450\text{pounds})$. How can you even lift your arm?

The reason, as FIGURE 14.8 shows, is that a fluid exerts pressure forces in all directions. There is a downward force of $\approx 2000\text{N}$ on your forearm, but the air underneath your arm exerts an upward force of the same magnitude. The net force is very close to zero. (To be accurate, there is a net upward force called the buoyant force. We’ll study buoyancy in Section 14.4. The buoyant force of the air is usually too small to notice.)

FIGURE 14.8 Pressure forces in a fluid push with equal strength in all directions.

But, you say, there isn’t any air under my arm if I rest it on a table. Actually, there is. There would be a vacuum under your arm if there were no air. Imagine placing your arm on the top of a large vacuum cleaner suction tube. What happens? You feel a downward force as the vacuum cleaner “tries to suck your arm in.” However, the downward force you feel is not a pulling force from the vacuum cleaner. It is the pushing force of the air above your arm when the air beneath your arm is removed and cannot push back. Air molecules do not have hooks! They have no ability to “pull” on your arm. The air can only push.

Vacuum cleaners, suction cups, and other similar devices are powerful examples of how strong atmospheric pressure forces can be if the air is removed from one side of an object so as to produce an unbalanced force. The fact that we are surrounded by the fluid allows us to move around in the air, just as we swim underwater, oblivious of these strong forces.

EXAMPLE 14.2 A suction cup

A 10.0-cm-diameter suction cup is pushed against a smooth ceiling. What is the maximum mass of an object that can be suspended from the suction cup without pulling it off the ceiling? The mass of the suction cup is negligible.

MODEL Pushing the suction cup against the ceiling pushes the air out. We’ll assume that the volume enclosed between the suction cup and the ceiling is a perfect vacuum with $p=0\text{Pa}$. We’ll also assume that the pressure in the room is 1 atm.

VISUALIZE FIGURE 14.9 shows a free-body diagram of the suction cup stuck to the ceiling. The downward normal force of the ceiling is distributed around the rim of the suction cup, but in the particle model we can show this as a single force vector.

FIGURE 14.9 A suction cup is held to the ceiling by air pressure pushing upward on the bottom.

FIGURE 14.9 Full Alternative Text

SOLVE The suction cup remains stuck to the ceiling, in equilibrium, as long as $F_{\text{air}}=n+F_{\text{G}}$. The magnitude of the upward force exerted by the air is

$$F_{\text{air}}=pA=p\pi r^2=(101\text{,}300\text{Pa})\pi (0.050\text{m}{)}^2=796\text{N}$$

There is no downward force from the air in this case because there is no air inside the cup. Increasing the hanging mass decreases the normal force $n$ by an equal amount. The maximum weight has been reached when $n$ is reduced to zero. Thus

$$\begin{array}{l}{(F_{\text{G}})}_{\text{max}}=mg=F_{\text{air}}=796\text{ N}\hfill \\ m={\displaystyle \frac{796\text{ N}}{g}}=81\text{ kg}\hfill \end{array}$$

REVIEW The suction cup can support a mass of up to $81\text{kg}$ if all the air is pushed out, leaving a perfect vacuum inside. A real suction cup won’t achieve a perfect vacuum, but suction cups can hold substantial weight.