Fluids and Pressure
Chapter Review
In this chapter we study fluids.
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Fluids are characterized by their ability to flow; both liquids and gases are considered to be fluids. An understanding of fluid behavior is essential to life and applications of this understanding are essential to many of the conveniences of modern living.
15-1 - 15-2 Density and Pressure
One of the more convenient properties used to describe a fluid is its density 
. The density of a substance is a measure of how compact the substance is; that is, how much mass is packed into a volume of the substance. On average the density of a substance is the amount of mass M divided by the volume V taken up by that mass
. The density of a substance is a measure of how compact the substance is; that is, how much mass is packed into a volume of the substance. On average the density of a substance is the amount of mass M divided by the volume V taken up by that mass.
Another important quantity in the study of fluids is pressure P. On the average, the pressure that is applied to an object is the amount of force F (normal to the surface) divided by the area A over which the force spreads
.
As the above expressions indicates, the units of pressure are those of force divided by area; in SI units this is called the pascal (Pa): 1 Pa = 1 N/m2. An important property of the pressure in a fluid is that it is equally applied in all directions (at a given depth) and applies forces that are perpendicular to any surface in the fluid.
When considering the pressure on or within a fluid it is important to recognize that for many applications we must account for a constant atmospheric pressure (Pat = 1.01 x 105 Pa). Because of the constant presence of the atmosphere we are often only interested in the pressure above and beyond that applied by the atmosphere. This additional pressure is called the gauge pressurePg
,
where P is the total pressure applied (sometimes called the absolute pressure).
Exercise 15.1 The Density of a Fluid: If 1.00 gallons of a certain fluid weighs 3.22 lb, what is its density?
Solution: We are given the following information:
Given: V = 1.00 gal, W = 3.22 lb Find:
Since the density is given by = m/V we need to find the mass from the weight and convert everything to SI units. To get the mass we have
= m/V we need to find the mass from the weight and convert everything to SI units. To get the mass we have.
The volume can be immediately converted to give
.
We are now ready to calculate the density as
.
Example 15.2 Gauge Pressure of Water: A uniform cylindrical container has a radius of 7.8 cm and a height of 13.2 cm. If this container is completely filled with water, what gauge pressure does the water apply to the bottom of the container?
Picture the Problem The picture shows a cylindrical container filled with water.
Strategy The pressure applied by the water will be the force that the water applies (equal to its weight) divided by the bottom area of the cylinder.
Solution
| 1. The volume of water equals the volume of the cylinder: |  |
| 2. Determine mass of the water |  |
| 3. Determine the weight of the water: |  |
| 4. Obtain the gauge pressure: |  |
Insight The above answer is the gauge pressure because we have ignored atmospheric pressure.
Practice Quiz
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15-3 Static Equilibrium in Fluids: Pressure and Depth
In this and the next several sections we will consider the properties of static fluids, that is, fluids that do not flow. In the case of static fluids, every part of the fluid and every object within the fluid is in static equilibrium. One of the basic properties of fluids that is very important to our ability to understand fluid behavior is known as Pascal's principle:
External pressure applied to an enclosed fluid is transmitted unchanged throughout the fluid.
Pascal's principle is important in determining the dependence of pressure on the depth within a fluid. Without any external pressure applied to the outer surface of a fluid, the pressure measured at a depth h beneath the surface arises from the weight of the fluid above the given level. The amount of this increase in pressure is given by gh. If there is external pressue on the fluid, such as from the atmosphere and/or any other source, this pressure is transmitted undiminished to every point in the fluid and it must be added to the pressure due to the weight of the fluid. Thus, for the dependence of pressure on depth we have
gh. If there is external pressue on the fluid, such as from the atmosphere and/or any other source, this pressure is transmitted undiminished to every point in the fluid and it must be added to the pressure due to the weight of the fluid. Thus, for the dependence of pressure on depth we have,
where P2 is the pressure at a given level within the fluid and P1 is the pressure at a height h above that level.
Example 15.3 Blood Pressure with Depth: Human blood has a density of approximately 1.05 x 103 kg/m3. Use this information to estimate the difference in blood pressure between the brain and the feet in a person who is approximately six feet tall.
Picture the Problem The picture shows a person approximately six feet in height.
Strategy We attempt this approximation to two significant figures by applying the above result for the dependence of pressure on depth using blood as the fluid.
Solution
| 1. Convert the height to meters: |  |
| 2. The difference in pressure is given by: |  |
| 3. Obtain the numerical result: |  |
Insight This is only an estimate for many reasons. The blood in the body is not a static fluid, but it flows; and between the head and the feet there is a pump (the heart) which will affect the result. If you plan to study medicine, see if you can find out the difference in blood pressure between the head and the feet.
Pascal's principle is also key to understanding the hydraulic lift. This device uses fluid pressure to convert a small input force into a large output force. The input force F1 is applied to a fluid over a comparatively small area A1 giving rise to a pressure change of F1/A1 = F2/A2, we must get a larger output force F2 > F1 if it is spread over a larger area A2 > A1. Therefore,
.
The consequence of getting this larger output force is that the distance through which this force can move an object at the output, d2, is smaller than the distance at the input d1. Since the same volume of fluid moves at the input and output, A1d1 = A2d2, the output distance is given by
.
Exercise 15.4 Hydraulic Lift: Your job at Dave's Manufacturing Company is to design a hydraulic lift for a client. This client typically needs to raise objects through a height of 0.500 meters. The system should be easily used by people of average height so the input distance should not exceed 5.00 ft. What would be a good ratio of output area to input area to consider?
Solution: We are given the following information.
Given: d1 = 5.00 ft, d2 = 0.500 m Find: A2/A1
From the information given, we know that the ratio of the input distance to the output distance is directly proportional to the ratio we seek
.
Therefore, a ratio of
The output area should be at least 3.05 times greater than the input area. Of course, in a more realistic situation you'd want more information from the client. What additional information might you want?
Practice Quiz
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