Pressure:
The pressure in a fluid due to a column of the fluid is:
where "rho" is the density of the fluid and h is its depth. This pressure, which is the force per unit area due to the fluid alone, is measured in Pascals (Pa).
For a diver submerged in a lake or the ocean, this pressure (called gauge pressure since it's what most gauges would read) is augmented by the pressure of the atmosphere. If the lake is at sea level and the barometric pressure is neither high nor low, the air pressure at sea level is 14.7 psi (pounds per square inch, 1 atm, or 1.103 x 105 Pa). The absolute pressure is the sum:
A characteristic of a fluid is that pressure is transmitted equally in all directions. Figure 15.1 on page 416 shows this principle being used: the pressure on the smaller cylinder (F1/A1) equals the pressure on the larger cylinder (F2/A2). Since the larger cylinder has a larger area, the fact that the ratios must be equal means that a large force can be exerted on the vehicle. Please study Example 15.1 on page 420 at this time. Do you see that Serway did not need to convert from cm to m when computing the force F1? (Of course, it was necessary in order to compute the pressure!)
Two devices for measuring pressure are shown on page 421, Figure 15.6; the manometer and barometer. These are rather simple instruments; meteorologists use considerably more sophisticated devices!
Forces on Submerged Sections:
Example 15.2, page 421 shows how the force on a dam may be computed. Since the pressure varies with depth, the force also varies and must be first computed on a differential section (across which the force is constant). Then these differential forces are summed (an integral). You should study this example, as well as my examples:
Example 1: A Gate in a DamArchimedes' Principle
Example 2: A Gate in a Dam - Revisited
A Bit of History, According to Joe:
Archimedes' lived in the Greek city-state of Syracuse. One day, the king ordered his goldsmith to make him a new crown. The king liked his new crown but he was suspicious, as kings often are. So he called in Archimedes and asked him how he'd like to keep his head. Archimedes' admitted that he was kinda attached to it, so the king told him: "Archie, old friend, find out if my new crown is gold or lead coated with gold, and I don't want it scratched! You've got a month or you lose your head!" (Naturally, I translated the local Greek lingo into idiomatic English.) So Archimedes took the crown & thought about it...And thought about it...And thought about it. One day he was taking his monthly bath (the one he took whether he needed it or not), when he jumped out of the bath running stark naked through the streets of Syracuse shouting "Eureka!" (Which supposedly means: "I've got it!")
Besides the fact that Archie was the first streaker, what Archimedes' "got" was that he realized that an object which sinks displaces its own volume of fluid. The weight of this fluid displaced is the buoyant force and is now known as "Archimedes' Principle":
It applies to an object which floats as well as one which sinks as long as you realize that the volume displaced does not necessarily equal the volume of the object. Serway proves the statement if you'd care to read it.
Archimedes was pretty bright. He invented the Archimedean Screw (no, that wasn't a device to get even with nasty kings). The purpose of the Screw was to raise water from a reservoir or stream to canals for irrigating crops. Third World countries still use it. We use it but call it a "snowblower".
Archimedes' fame was known even to the Romans. When they attacked the city (several times), Archimedes invented a hoist which grabbed the Roman ships, lifted them high into the air, and dashed them on the rocks. The Roman commander gave orders that Archimedes was to be taken alive - they wanted him for his engineering expertise! When the Romans finally broke into the city, a legionary supposedly found Archimedes in his laboratory. "Come with me!", he ordered. Ever the absorbed researcher, Archimedes replied: "Just a minute." At which the annoyed Roman ran him through with his spear. History does not record what happened to the legionary! (This sequence of events is hardly history and is probably apochryphal.)
Specific Gravity and Density:
The SI units for density are kg/m3. (This used to be called "mass density".) Water's density is 1000 kg/m3. Another commonly used system of units is the cgs (centimeter-gram-second); in this, water's density is 1.00 g/cm3. Specific gravity (s.g.) is the ratio of the density of the substance to that of water. For example, granite, which has a density of about 2.70 g/cm3, has a specific gravity of 2.70 (no units, since it's a ratio). In U.S. Customary units, the density of water (it's a weight density, not a mass density and an upper case D is used to represent it) is 62.4 lbs/ft3.
Note that, if you're given specific gravity, it's numerically the same in cgs. Multiply it by 1000 to get SI. Multiply the s.g. by 62.4 to get weight density in US Customary units.
(Comment: "Specific" quantities are often used in chemical engineering. Specific volume is m3/kg - the reciprocal of density. Specific enthalpy is given in kJ/kg or kJ/mol, and specific internal energy is also measured in kJ/kg or kJ/mol.)
Example 3: Apparent weight
Example 4: A Kickboard
Fluid Dynamics
The Equation of Continuity:
This is essentially a volume balance; i.e., the volume entering must equal the volume leaving. Since pipes do not usually blow up like balloons, this seems to be a pretty logical statement. However, the fluid that this statement applies to must be "incompressible". Now that's not true about anything; one can change the volume of any material is enough pressure is applied! Even atomic nuclei are compressible under the fantastic pressures of neutron stars! Like most everything real, we are working with an approximation. For the pressures used, the compressibility (incidentially, compressibility is defined as the reciprocal of the bulk modulus so you'd have to have an infinite modulus to get a compressibility of zero) causes negligible changes in the volume.
Look at Figure 15.13 on page 427. As the fluid moves through a short initial distance (delta-x1), if this distance is short, the input cross sectional area is constant and the volumetric flow rate is A1v1 m3/s. If the pipe is not flexible and the fluid is essentially incompressible (no gases), the same relationship holds at the other end of the pipe:
Since incompressibility implies constant density, we could use a mass balance instead of a volume balance if that's easier.
Example 5: Emptying a cellar
Bernoulli's Equation:
Please look at Figure 15.14 on page 428. At first sight, this may seem to be a repeat of the above. But the difference is that we're now considering elevation changes and pressure differences. Serway does the derivation, which is merely conservation of energy (he uses the work-energy theorem and I would write it as an energy balance but it's the same thing).
If a small amount of mass of fluid is forced through the pipe,
The volumes do not have subscripts since the same volume exits as enters; the fluid is incompressible. For the same reason, the densities are the same.
The other statement you might have a problem with is that pressure times volumetric change is the work done. Consider a piston of cross sectional area A moving through a distance x:
The volume it moves though (compresses) is Ax. The pressure is F/A. Let's look at the work done:
You should follow through Serway's derivation of Bernoulli's Principle. Here's an application:
Example 6: The cellar revisited
Applications of Bernoulli's Principle:
Serway has a good example which you should study - Example 15.5 on page 429 (The Venturi Tube). You might want to use another reference to look up Torricelli's Law.