13.4 - Pressure and fluids
As a submarine dives deeper and deeper into the
sea, it encounters increasing pressure. The nightmare
of any submariner is that his craft goes so deep it
collapses under the tremendous pressure of the
water. (Rent the movie U-571 if you want to watch a
Hollywood thriller that deals with pressure at great
depths.)
Physicists prefer a little less drama and a little more
measurement in their dealings with pressure. They
have developed an equation to describe the pressure
of a fluid as a function of the fluid’s depth.
Why does water pressure increase as a submarine
descends? The pressure increases because the amount of water on top of the submarine increases as the vessel does down. The weight of
the water exerts a force over the entire surface area of the submarine’s hull. The water pressure does not depend on the orientation of the
surface. It exists all over the craft: on its top, bottom and sides.
The first equation on the right shows how to calculate the pressure at a point underneath the surface of a fluid. The pressure equals the
product of the fluid’s density, the constant acceleration of gravity, and the height of the column of fluid above the point. “Height” means the
distance from the point to the surface of the fluid. The equation states that pressure increases linearly with depth. The water pressure at 200
meters down is twice as great as it is at 100 meters. This equation applies when the fluid is static.
The fluid pressure varies solely as a function of the density ȡ and the depth in the fluid, not with the shape of the container holding the fluid. If
you fill a swimming pool and a Coke bottle with water, the water pressure at 0.1 meters below the surface will be the same in both containers.
The pressure will also be the same at the bottom of the bottle and on the wall of the pool at the same depth. Whether the surface area is
horizontal or vertical, the pressure is the same.
You can add pressures. The total pressure exerted on the exterior of the hull of the submarine equals the sum of atmospheric pressure (the
pressure exerted by the Earth’s air above it) and the pressure of the water above it. The two pressures must be calculated separately and
added as shown in Equation 2. Since the densities of water and air differ, the pressure of the combined column of fluids above the submarine
cannot be calculated as a single product ȡgh.
We have implicitly described two types of pressure: absolute and gauge pressure. The total pressure is called the absolute pressure. It is the
sum of the atmospheric pressure and the pressure of the fluid in question, in this case water. The photograph below shows how a Styrofoam®
cup (as shown on the right) was crushed (as shown on the left) by an absolute pressure of 3288 psi when it was submerged to a depth of more
than two kilometers below the ocean’s surface.
The term gauge pressure describes the pressure caused solely by the water (or other fluid), ignoring the atmospheric pressure. The gauge
pressure equals ȡgh, where ȡ,g and h are measured for the fluid alone. To state the same concept in another way: Gauge pressure equals
the absolute pressure minus the atmospheric pressure.
Pressures can oppose one other, when they act on opposite sides of the same surface. For example, the “atmospheric” pressure inside an
airplane cabin is allowed to decrease as the plane climbs. The pressure inside your ear can be higher than the pressure of the cabin, since it
may remain at the higher pressure of the atmosphere at the Earth’s surface. In this case, the pressure in parts of your ears is greater than the
pressure outside them, producing a net outward pressure (and force) on your eardrums. The result is that your ear begins to ache.
You can reduce the pressure inside your ears by chewing gum or yawning to “pop” them. This stretches and opens the Eustachian tubes,
passages between your ears and throat. Air flows out of your ear into the cabin, balancing the pressures, and your earache disappears.
If you look at the large value calculated in the example problem for the absolute pressure on the inner surface of the aquarium’s bottom, you
might wonder why the tank does not burst. Remember that atmospheric pressure is pushing inward on its exterior surfaces as well. The net
pressure on the bottom plate is the gauge pressure due solely to the water, which is not large enough to cause the tank to burst.
The density of a liquid does not vary significantly with depth, so using an average density figure in the formula ȡgh provides a good
approximation of the pressure at any depth. The density of gases can vary greatly, so the formula is not as applicable to them, especially if h is
Navy personnel performing under pressure.