College Physics for AP® Courses

# 11.6Gauge Pressure, Absolute Pressure, and Pressure Measurement

College Physics for AP® Courses11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement

### Learning Objectives

By the end of this section, you will be able to:

• Define gauge pressure and absolute pressure.
• Understand the working of aneroid and open-tube barometers.

If you limp into a gas station with a nearly flat tire, you will notice the tire gauge on the airline reads nearly zero when you begin to fill it. In fact, if there were a gaping hole in your tire, the gauge would read zero, even though atmospheric pressure exists in the tire. Why does the gauge read zero? There is no mystery here. Tire gauges are simply designed to read zero at atmospheric pressure and positive when pressure is greater than atmospheric.

Similarly, atmospheric pressure adds to blood pressure in every part of the circulatory system. (As noted in Pascal's Principle, the total pressure in a fluid is the sum of the pressures from different sources—here, the heart and the atmosphere.) But atmospheric pressure has no net effect on blood flow since it adds to the pressure coming out of the heart and going back into it, too. What is important is how much greater blood pressure is than atmospheric pressure. Blood pressure measurements, like tire pressures, are thus made relative to atmospheric pressure.

In brief, it is very common for pressure gauges to ignore atmospheric pressure—that is, to read zero at atmospheric pressure. We therefore define gauge pressure to be the pressure relative to atmospheric pressure. Gauge pressure is positive for pressures above atmospheric pressure, and negative for pressures below it.

### Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. Gauge pressure is positive for pressures above atmospheric pressure, and negative for pressures below it.

In fact, atmospheric pressure does add to the pressure in any fluid not enclosed in a rigid container. This happens because of Pascal's principle. The total pressure, or absolute pressure, is thus the sum of gauge pressure and atmospheric pressure: $Pabs=Pg+PatmPabs=Pg+Patm size 12{P rSub { size 8{"abs"} } =P rSub { size 8{g} } +P rSub { size 8{"atm"} } } {}$ where $PabsPabs size 12{P rSub { size 8{"abs"} } } {}$ is absolute pressure, $PgPg size 12{P rSub { size 8{g} } } {}$ is gauge pressure, and $PatmPatm size 12{P rSub { size 8{"atm"} } } {}$ is atmospheric pressure. For example, if your tire gauge reads 34 psi (pounds per square inch), then the absolute pressure is 34 psi plus 14.7 psi ($PatmPatm size 12{P rSub { size 8{"atm"} } } {}$ in psi), or 48.7 psi (equivalent to 336 kPa).

### Absolute Pressure

Absolute pressure is the sum of gauge pressure and atmospheric pressure.

For reasons we will explore later, in most cases the absolute pressure in fluids cannot be negative. Fluids push rather than pull, so the smallest absolute pressure is zero. (A negative absolute pressure is a pull.) Thus the smallest possible gauge pressure is $Pg=−PatmPg=−Patm size 12{P rSub { size 8{g} } = - P rSub { size 8{"atm"} } } {}$ (this makes $PabsPabs size 12{P rSub { size 8{"abs"} } } {}$ zero). There is no theoretical limit to how large a gauge pressure can be.

There are a host of devices for measuring pressure, ranging from tire gauges to blood pressure cuffs. Pascal's principle is of major importance in these devices. The undiminished transmission of pressure through a fluid allows precise remote sensing of pressures. Remote sensing is often more convenient than putting a measuring device into a system, such as a person's artery.

Figure 11.17 shows one of the many types of mechanical pressure gauges in use today. In all mechanical pressure gauges, pressure results in a force that is converted (or transduced) into some type of readout.

Figure 11.17 This aneroid gauge utilizes flexible bellows connected to a mechanical indicator to measure pressure.

An entire class of gauges uses the property that pressure due to the weight of a fluid is given by $P=hρg.P=hρg. size 12{P=hρg "." } {}$ Consider the U-shaped tube shown in Figure 11.18, for example. This simple tube is called a manometer. In Figure 11.18(a), both sides of the tube are open to the atmosphere. Atmospheric pressure therefore pushes down on each side equally so its effect cancels. If the fluid is deeper on one side, there is a greater pressure on the deeper side, and the fluid flows away from that side until the depths are equal.

Let us examine how a manometer is used to measure pressure. Suppose one side of the U-tube is connected to some source of pressure $PabsPabs size 12{P rSub { size 8{"abs"} } } {}$ such as the toy balloon in Figure 11.18(b) or the vacuum-packed peanut jar shown in Figure 11.18(c). Pressure is transmitted undiminished to the manometer, and the fluid levels are no longer equal. In Figure 11.18(b), $PabsPabs size 12{P rSub { size 8{"abs"} } } {}$ is greater than atmospheric pressure, whereas in Figure 11.18(c), $PabsPabs size 12{P rSub { size 8{"abs"} } } {}$ is less than atmospheric pressure. In both cases, $PabsPabs size 12{P rSub { size 8{"abs"} } } {}$ differs from atmospheric pressure by an amount $hρghρg size 12{hρg} {}$, where $ρρ size 12{ρ} {}$ is the density of the fluid in the manometer. In Figure 11.18(b), $PabsPabs size 12{P rSub { size 8{"abs"} } } {}$ can support a column of fluid of height $hh size 12{h} {}$, and so it must exert a pressure $hρghρg size 12{hρg} {}$ greater than atmospheric pressure (the gauge pressure $PgPg size 12{P rSub { size 8{g} } } {}$ is positive). In Figure 11.18(c), atmospheric pressure can support a column of fluid of height $hh size 12{h} {}$, and so $PabsPabs size 12{P rSub { size 8{"abs"} } } {}$ is less than atmospheric pressure by an amount $hρghρg size 12{hρg} {}$ (the gauge pressure $PgPg size 12{P rSub { size 8{g} } } {}$ is negative). A manometer with one side open to the atmosphere is an ideal device for measuring gauge pressures. The gauge pressure is $Pg=hρgPg=hρg size 12{P rSub { size 8{g} } =hρg} {}$ and is found by measuring $hh size 12{h} {}$.

Figure 11.18 An open-tube manometer has one side open to the atmosphere. (a) Fluid depth must be the same on both sides, or the pressure each side exerts at the bottom will be unequal and there will be flow from the deeper side. (b) A positive gauge pressure $Pg=hρgPg=hρg size 12{P rSub { size 8{g} } =hρg} {}$ transmitted to one side of the manometer can support a column of fluid of height $hh size 12{h} {}$. (c) Similarly, atmospheric pressure is greater than a negative gauge pressure $PgPg size 12{P rSub { size 8{g} } } {}$ by an amount $hρghρg size 12{hρg} {}$. The jar's rigidity prevents atmospheric pressure from being transmitted to the peanuts.

Mercury manometers are often used to measure arterial blood pressure. An inflatable cuff is placed on the upper arm as shown in Figure 11.19. By squeezing the bulb, the person making the measurement exerts pressure, which is transmitted undiminished to both the main artery in the arm and the manometer. When this applied pressure exceeds blood pressure, blood flow below the cuff is cut off. The person making the measurement then slowly lowers the applied pressure and listens for blood flow to resume. Blood pressure pulsates because of the pumping action of the heart, reaching a maximum, called systolic pressure, and a minimum, called diastolic pressure, with each heartbeat. Systolic pressure is measured by noting the value of $hh size 12{h} {}$ when blood flow first begins as cuff pressure is lowered. Diastolic pressure is measured by noting $hh size 12{h} {}$ when blood flows without interruption. The typical blood pressure of a young adult raises the mercury to a height of 120 mm at systolic and 80 mm at diastolic. This is commonly quoted as 120 over 80, or 120/80. The first pressure is representative of the maximum output of the heart; the second is due to the elasticity of the arteries in maintaining the pressure between beats. The density of the mercury fluid in the manometer is 13.6 times greater than water, so the height of the fluid will be 1/13.6 of that in a water manometer. This reduced height can make measurements difficult, so mercury manometers are used to measure larger pressures, such as blood pressure. The density of mercury is such that $1.0 mm Hg=133Pa1.0 mm Hg=133Pa$.

### Systolic Pressure

Systolic pressure is the maximum blood pressure.

### Diastolic Pressure

Diastolic pressure is the minimum blood pressure.

Figure 11.19 In routine blood pressure measurements, an inflatable cuff is placed on the upper arm at the same level as the heart. Blood flow is detected just below the cuff, and corresponding pressures are transmitted to a mercury-filled manometer. (credit: U.S. Army photo by Spc. Micah E. Clare\4TH BCT)

### Example 11.7

#### Calculating Height of IV Bag: Blood Pressure and Intravenous Infusions

Intravenous infusions are usually made with the help of the gravitational force. Assuming that the density of the fluid being administered is 1.00 g/ml, at what height should the IV bag be placed above the entry point so that the fluid just enters the vein if the blood pressure in the vein is 18 mm Hg above atmospheric pressure? Assume that the IV bag is collapsible.

#### Strategy for (a)

For the fluid to just enter the vein, its pressure at entry must exceed the blood pressure in the vein (18 mm Hg above atmospheric pressure). We therefore need to find the height of fluid that corresponds to this gauge pressure.

#### Solution

We first need to convert the pressure into SI units. Since $1.0 mm Hg=133 Pa1.0 mm Hg=133 Pa$,

$P = 18 mm Hg × 133 Pa 1.0 mm Hg = 2400 Pa . P = 18 mm Hg × 133 Pa 1.0 mm Hg = 2400 Pa . size 12{P="18""mm""Hg" times { {"133""Pa"} over {1 "." 0"mm""Hg"} } ="2400""Pa" "." } {}$
11.28

Rearranging $Pg=hρgPg=hρg size 12{P rSub { size 8{g} } =hρg} {}$ for $hh size 12{h} {}$ gives $h=Pgρgh=Pgρg size 12{h= { {P rSub { size 8{g} } } over {ρg} } } {}$. Substituting known values into this equation gives

h = 2400 N /m 2 1 . 0 × 10 3 kg/m 3 9 . 80 m/s 2 = 0.24 m. h = 2400 N /m 2 1 . 0 × 10 3 kg/m 3 9 . 80 m/s 2 = 0.24 m. alignl { stack { size 12{h= { {"2400""N/m" rSup { size 8{2} } } over { left (1 "." 0 times "10" rSup { size 8{3} } "kg/m" rSup { size 8{3} } right ) left (9 "." "80""m/s" rSup { size 8{2} } right )} } } {} # " "=" 0" "." "24"m "." {} } } {}
11.29

#### Discussion

The IV bag must be placed at 0.24 m above the entry point into the arm for the fluid to just enter the arm. Generally, IV bags are placed higher than this. You may have noticed that the bags used for blood collection are placed below the donor to allow blood to flow easily from the arm to the bag, which is the opposite direction of flow than required in the example presented here.

A barometer is a device that measures atmospheric pressure. A mercury barometer is shown in Figure 11.20. This device measures atmospheric pressure, rather than gauge pressure, because there is a nearly pure vacuum above the mercury in the tube. The height of the mercury is such that $hρg=Patmhρg=Patm size 12{hρg=P rSub { size 8{"atm"} } } {}$. When atmospheric pressure varies, the mercury rises or falls, giving important clues to weather forecasters. The barometer can also be used as an altimeter, since average atmospheric pressure varies with altitude. Mercury barometers and manometers are so common that units of mm Hg are often quoted for atmospheric pressure and blood pressures. Table 11.2 gives conversion factors for some of the more commonly used units of pressure.

Figure 11.20 A mercury barometer measures atmospheric pressure. The pressure due to the mercury's weight, $hρghρg size 12{hρg} {}$, equals atmospheric pressure. The atmosphere is able to force mercury in the tube to a height $hh size 12{h} {}$ because the pressure above the mercury is zero.
Conversion to N/m2 (Pa) Conversion from atm
$1.0 atm = 1 . 013 × 10 5 N/m 2 1.0 atm = 1 . 013 × 10 5 N/m 2 size 12{1 "." 0"atm"=1 "." "013" times "10" rSup { size 8{5} } "N/m" rSup { size 8{2} } } {}$ $1.0 atm = 1 . 013 × 10 5 N/m 2 1.0 atm = 1 . 013 × 10 5 N/m 2 size 12{1 "." 0"atm"=1 "." "013" times "10" rSup { size 8{5} } "N/m" rSup { size 8{2} } } {}$
$1.0 dyne/cm 2 = 0 . 10 N/m 2 1.0 dyne/cm 2 = 0 . 10 N/m 2 size 12{1 "." 0"dyne/cm" rSup { size 8{2} } =0 "." "10""N/m" rSup { size 8{2} } } {}$ $1 . 0 atm = 1 . 013 × 10 6 dyne/cm 2 1 . 0 atm = 1 . 013 × 10 6 dyne/cm 2 size 12{1 "." 0"atm"=1 "." "013" times "10" rSup { size 8{6} } "dyne/cm" rSup { size 8{2} } } {}$
$1 . 0 kg/cm 2 = 9 . 8 × 10 4 N/m 2 1 . 0 kg/cm 2 = 9 . 8 × 10 4 N/m 2 size 12{1 "." 0"kg/cm" rSup { size 8{2} } =9 "." 8 times "10" rSup { size 8{4} } "N/m" rSup { size 8{2} } } {}$ $1 . 0 atm = 1 . 013 kg/cm 2 1 . 0 atm = 1 . 013 kg/cm 2 size 12{1 "." 0"atm"=1 "." "013""kg/cm" rSup { size 8{2} } } {}$
$1 . 0 lb/in . 2 = 6 . 90 × 10 3 N/m 2 1 . 0 lb/in . 2 = 6 . 90 × 10 3 N/m 2 size 12{1 "." 0"lb/in" "." rSup { size 8{2} } =6 "." "90" times "10" rSup { size 8{3} } "N/m" rSup { size 8{2} } } {}$ $1 . 0 atm = 14 . 7 lb/in . 2 1 . 0 atm = 14 . 7 lb/in . 2 size 12{1 "." 0"atm"="14" "." 7"lb/in" "." rSup { size 8{2} } } {}$
$1.0 mm Hg = 133 N/m 2 1.0 mm Hg = 133 N/m 2 size 12{1 "." 0"mm""Hg"="133""N/m" rSup { size 8{2} } } {}$ $1 . 0 atm = 760 mm Hg 1 . 0 atm = 760 mm Hg size 12{1 "." 0"atm"="760""mm""Hg"} {}$
$1 . 0 cm Hg = 1 . 33 × 10 3 N/m 2 1 . 0 cm Hg = 1 . 33 × 10 3 N/m 2 size 12{1 "." 0"cm""Hg"=1 "." "33" times "10" rSup { size 8{3} } "N/m" rSup { size 8{2} } } {}$ $1 . 0 atm = 76 . 0 cm Hg 1 . 0 atm = 76 . 0 cm Hg size 12{1 "." 0"atm"="76" "." 0"cm""Hg"} {}$
$1 . 0 cm water = 98 . 1 N/m 2 1 . 0 cm water = 98 . 1 N/m 2 size 12{1 "." 0"cm""water"="98" "." 1"N/m" rSup { size 8{2} } } {}$ $1 . 0 atm = 1 . 03 × 10 3 cm water 1 . 0 atm = 1 . 03 × 10 3 cm water size 12{1 "." 0"atm"=1 "." "03" times "10" rSup { size 8{3} } "cm""water"} {}$
$1.0 bar = 1 . 000 × 10 5 N/m 2 1.0 bar = 1 . 000 × 10 5 N/m 2 size 12{1 "." 0"bar"=1 "." "000" times "10" rSup { size 8{5} } "N/m" rSup { size 8{2} } } {}$ $1 . 0 atm = 1.013 bar 1 . 0 atm = 1.013 bar size 12{1 "." 0"atm"=1 "." "013""bar"} {}$
$1.0 millibar = 1 . 000 × 10 2 N/m 2 1.0 millibar = 1 . 000 × 10 2 N/m 2 size 12{1 "." 0"millibar"=1 "." "000" times "10" rSup { size 8{2} } "N/m" rSup { size 8{2} } } {}$ $1.0 atm = 1013 millibar 1.0 atm = 1013 millibar$
Table 11.2 Conversion Factors for Various Pressure Units
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