14.2 Measuring Pressure

Manometers

One of the most important classes of pressure gauges applies the property that pressure due to the weight of a fluid of constant density is given by [latex]p=h\rho g[/latex]. The U-shaped tube shown in Figure is an example of a manometer; in part (a), both sides of the tube are open to the atmosphere, allowing atmospheric pressure to push down on each side equally so that its effects cancel.

A manometer with only one side open to the atmosphere is an ideal device for measuring gauge pressures. The gauge pressure is [latex]{p}_{\text{g}}=h\rho g[/latex] and is found by measuring h. For example, suppose one side of the U-tube is connected to some source of pressure [latex]{p}_{\text{abs}},[/latex] such as the balloon in part (b) of the figure or the vacuum-packed peanut jar shown in part (c). Pressure is transmitted undiminished to the manometer, and the fluid levels are no longer equal. In part (b), [latex]{p}_{\text{abs}}[/latex] is greater than atmospheric pressure, whereas in part (c), [latex]{p}_{\text{abs}}[/latex] is less than atmospheric pressure. In both cases, [latex]{p}_{\text{abs}}[/latex] differs from atmospheric pressure by an amount [latex]h\rho g,[/latex] where [latex]\rho[/latex] is the density of the fluid in the manometer. In part (b), [latex]{p}_{\text{abs}}[/latex] can support a column of fluid of height h, so it must exert a pressure [latex]h\rho g[/latex] greater than atmospheric pressure (the gauge pressure [latex]{p}_{\text{g}}[/latex] is positive). In part (c), atmospheric pressure can support a column of fluid of height h, so [latex]{p}_{\text{abs}}[/latex] is less than atmospheric pressure by an amount [latex]h\rho g[/latex] (the gauge pressure [latex]{p}_{\text{g}}[/latex] is negative).

Barometers

Manometers typically use a U-shaped tube of a fluid (often mercury) to measure pressure. A barometer (see Figure) is a device that typically uses a single column of mercury to measure atmospheric pressure. The barometer, invented by the Italian mathematician and physicist Evangelista Torricelli (1608–1647) in 1643, is constructed from a glass tube closed at one end and filled with mercury. The tube is then inverted and placed in a pool of mercury. 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 [latex]h\rho g={p}_{\text{atm}}[/latex]. When atmospheric pressure varies, the mercury rises or falls.

Weather forecasters closely monitor changes in atmospheric pressure (often reported as barometric pressure), as rising mercury typically signals improving weather and falling mercury indicates deteriorating weather. 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.

Example

Fluid Heights in an Open U-Tube

A U-tube with both ends open is filled with a liquid of density [latex]{\rho }_{1}[/latex] to a height h on both sides (Figure). A liquid of density [latex]{\rho }_{2} \lt {\rho }_{1}[/latex] is poured into one side and Liquid 2 settles on top of Liquid 1. The heights on the two sides are different. The height to the top of Liquid 2 from the interface is [latex]{h}_{2}[/latex] and the height to the top of Liquid 1 from the level of the interface is [latex]{h}_{1}[/latex]. Derive a formula for the height difference.

Strategy

The pressure at points at the same height on the two sides of a U-tube must be the same as long as the two points are in the same liquid. Therefore, we consider two points at the same level in the two arms of the tube: One point is the interface on the side of the Liquid 2 and the other is a point in the arm with Liquid 1 that is at the same level as the interface in the other arm. The pressure at each point is due to atmospheric pressure plus the weight of the liquid above it.

[latex]\begin{array}{c}\text{Pressure on the side with Liquid 1}={p}_{0}+{\rho }_{1}g{h}_{1}\hfill \\ \text{Pressure on the side with Liquid 2}={p}_{0}+{\rho }_{2}g{h}_{2}\hfill \end{array}[/latex]

Solution

Since the two points are in Liquid 1 and are at the same height, the pressure at the two points must be the same. Therefore, we have

[latex]{p}_{0}+{\rho }_{1}g{h}_{1}={p}_{0}+{\rho }_{2}g{h}_{2}.[/latex]

Hence,

[latex]{\rho }_{1}{h}_{1}={\rho }_{2}{h}_{2}.[/latex]

This means that the difference in heights on the two sides of the U-tube is

[latex]{h}_{2}-{h}_{1}=(1-\frac{{p}_{1}}{{p}_{2}}){h}_{2}.[/latex]

The result makes sense if we set [latex]{p}_{2}={p}_{1},[/latex] which gives [latex]{h}_{2}={h}_{1}.[/latex] If the two sides have the same density, they have the same height.

Units of pressure

As stated earlier, the SI unit for pressure is the pascal (Pa), where

In addition to the pascal, many other units for pressure are in common use (Figure). In meteorology, atmospheric pressure is often described in the unit of millibars (mbar), where

The millibar is a convenient unit for meteorologists because the average atmospheric pressure at sea level on Earth is [latex]1.013\times {10}^{5}\,\text{Pa}=1013\,\text{mbar}=1\,\text{atm}[/latex]. Using the equations derived when considering pressure at a depth in a fluid, pressure can also be measured as millimeters or inches of mercury. The pressure at the bottom of a 760-mm column of mercury at [latex]0\,{}^\circ \text{C}[/latex] in a container where the top part is evacuated is equal to the atmospheric pressure. Thus, 760 mm Hg is also used in place of 1 atmosphere of pressure. In vacuum physics labs, scientists often use another unit called the torr, named after Torricelli, who, as we have just seen, invented the mercury manometer for measuring pressure. One torr is equal to a pressure of 1 mm Hg.