An auto-rickshaw driver stops at a petrol pump to check his tyres. The gauge reads about \(30\) psi in a good tyre. On the spare, which has gone flat, it reads zero. Yet the flat tyre is not empty: it is still full of air.
The air is a fluid too
We live at the bottom of an ocean of air, many kilometres deep. Air has weight, like any fluid, and the air above presses on everything below it. The pressure of the air at a place is the atmospheric pressure \(P_0\). At sea level it is about \(1.013 \times 10^5\,\text{Pa}\). In this chapter we round it to \(10^5\,\text{Pa}\).
That is a large pressure. On the palm of your hand, about \(8\,\text{cm}\) by \(10\,\text{cm}\), the air presses with about \(800\,\text{N}\). We do not feel it, because the air presses on the back of the hand just as hard. The fluids inside our bodies push outward with the same pressure too. Only a difference in pressure is felt.
Absolute and gauge pressure
A pressure can be measured from two zeros. The absolute pressure is measured from a perfect vacuum, where the pressure is truly zero. The gauge pressure is measured from the atmosphere: it is the excess, \(P - P_0\). Most gauges read gauge pressure, because they compare the pressure inside with the air outside.
The gauge pressure at a point is \(P_g = P - P_0\), the amount by which the absolute pressure \(P\) exceeds the atmospheric pressure \(P_0\). It is negative when the pressure is below atmospheric.
Now the petrol pump. A good tyre reading \(30\,\text{psi}\) holds air at about \(2 \times 10^5\,\text{Pa}\) above the atmosphere. Its absolute pressure is \(2 \times 10^5 + 10^5 = 3 \times 10^5\,\text{Pa}\). A flat tyre holds air at the same pressure as the air outside. Its gauge pressure is zero, so the gauge reads zero, although the tyre is full of air.
A gauge on a gas cylinder reads \(1.5 \times 10^5\,\text{Pa}\). Atmospheric pressure is \(10^5\,\text{Pa}\). What is the absolute pressure of the gas, in Pa?
Torricelli's tube
Fill a glass tube about a metre long with mercury, and close its open end with a finger. Turn it upside down into a dish of mercury, and take the finger away. The mercury in the tube falls, then stops with its top about \(76\,\text{cm}\) above the surface in the dish. Above the mercury, at the closed top of the tube, there is nothing but a vacuum. This is a mercury barometer.
Compare the points A and B of Fig 3.1. A is inside the tube at the level of the surface in the dish, and B is on that surface. They are at the same level in one connected liquid, so their pressures are equal. The pressure at B is \(P_0\), the atmosphere's. The pressure at A is that of the vacuum, zero, plus the column above it, \(\rho g h\). So \(P_0 = \rho g h\).
With the numbers of this chapter, \(P_0 = 13\,600 \times 10 \times 0.76 = 103\,360\,\text{Pa}\), about \(1.03 \times 10^5\,\text{Pa}\). The true value of \(g\) is closer to \(9.8\,\text{m/s}^2\), and then the same column gives \(1.013 \times 10^5\,\text{Pa}\). That is the standard atmosphere. In problems, use \(g = 10\) unless you are told otherwise, and expect the small difference.
The height in \(P_0 = \rho g h\) is the vertical height of the column. Tilt the tube, and more mercury flows in, so the column grows longer along the tube. Its top stays at the same height, \(76\,\text{cm}\) above the dish.
Why mercury and not water
Water would work, but the column would be very tall. Water is \(13.6\) times less dense than mercury, so it needs a column \(13.6\) times taller to press as hard. That is \(0.76 \times 13.6 = 10.3\,\text{m}\), taller than a three-storey house. Mercury's great density keeps the barometer about a metre long.
Higher up, there is less air above, so the column is shorter. At a hill station about \(2200\,\text{m}\) up, the second chip of Fig 3.1, it stands only about \(58\,\text{cm}\). The barometer measures the weight of the air above it.
On a stormy day a mercury barometer reads \(70\,\text{cm}\). Take \(g = 10\,\text{m/s}^2\) and the density of mercury as \(13\,600\,\text{kg/m}^3\). What is the atmospheric pressure, in Pa?
A barometer tube reads \(76\,\text{cm}\) when vertical. It is tilted to \(30°\) from the vertical, and its top is still above the mercury. What happens to the mercury in it?
In 1643 Evangelista Torricelli, a student of Galileo, filled a glass tube with mercury and turned it into a dish. The mercury stood about three quarters of a metre high, with empty space above. He wrote that "we live submerged at the bottom of an ocean of air". Five years later, Blaise Pascal had the experiment carried up the Puy de Dôme mountain in France. At the top the column was about \(8\,\text{cm}\) shorter, as the weight of less air above would make it.
The open-tube manometer
A manometer measures the pressure of a gas. It is a U-tube holding a liquid, often mercury. One arm is joined to the gas; the other is open to the air. If the gas pressure is larger than the atmosphere's, the gas pushes the liquid down in its arm and up in the open arm. The difference \(h\) between the two levels measures the difference in pressure.
To read a manometer, pick the level of the lower mercury surface, the dashed line in Fig 3.2. At that level the two arms have equal pressure, since mercury joins them. In the gas arm the pressure there is \(P_{\text{gas}}\). In the open arm it is the atmosphere plus the column above, \(P_0 + \rho g h\). So \(P_{\text{gas}} = P_0 + \rho g h\).
With the open arm \(20\,\text{cm}\) higher, \(\rho g h = 13\,600 \times 10 \times 0.20 = 27\,200\,\text{Pa}\). So \(P_{\text{gas}} = 1.272 \times 10^5\,\text{Pa}\), and the gauge pressure of the gas is \(27\,200\,\text{Pa}\). When the open arm is lower instead, the gas pushes less than the air. Then \(P_{\text{gas}} = P_0 - \rho g h = 0.728 \times 10^5\,\text{Pa}\), below atmospheric.
For small pressures a water manometer is better, since water is so much lighter. A household gas regulator, for example, delivers cooking gas at about \(30\,\text{cm}\) of water above the atmosphere. That is a gauge pressure of about \(3000\,\text{Pa}\), which would move mercury by only about \(2\,\text{cm}\).
In a mercury manometer, the open arm stands \(5\,\text{cm}\) lower than the arm joined to a gas. Take \(g = 10\,\text{m/s}^2\), mercury \(13\,600\,\text{kg/m}^3\) and \(P_0 = 10^5\,\text{Pa}\). Find the gas pressure, in Pa.
Two liquids in one U-tube
Pour oil into one arm of a U-tube that holds water. The oil floats on the water and pushes it down in that arm. Where should the pressures be compared? Only at a level with the same liquid below it in both arms: the level of the oil-water boundary. Below that level there is only water, joined through the bend.
At the dashed line of Fig 3.3, the left arm has \(20\,\text{cm}\) of oil above it, and the right arm \(16\,\text{cm}\) of water. Both columns are open to the air. So \(P_0 + \rho_{\text{oil}} g \times 0.20 = P_0 + 1000 \times g \times 0.16\). The atmosphere and \(g\) cancel, leaving \(\rho_{\text{oil}} = 1000 \times 16/20 = 800\,\text{kg/m}^3\). The lighter liquid needs the taller column.
Look at the U-tube of Fig 3.3. A student compares the pressure at the top of the water in the right arm with the pressure at that height in the oil. Why is this wrong?
A pump does not pull
Drink through a straw, and the juice seems to be pulled up. But a fluid cannot be pulled; it can only be pushed. Your mouth lowers the pressure at the top of the straw. The atmosphere, pressing on the juice in the glass, pushes it up the straw into the region of lower pressure.
A suction hand pump on a well works the same way. Its piston rises and lowers the pressure at the top of the pipe. The atmosphere pushes the water up the pipe, until the column is tall enough to balance the difference. For water to stand \(h\) above the well, the pressure at the top of the pipe must fall to \(P_0 - \rho g h\). For a well \(7\,\text{m}\) deep, that is \(10^5 - 7 \times 10^4 = 3 \times 10^4\,\text{Pa}\).
The best a suction pump can do is a perfect vacuum at the top. Then the column that the atmosphere holds up is \(h = P_0/\rho g = 10^5/10^4 = 10\,\text{m}\), about \(10.3\,\text{m}\) with the true values. It is the water barometer again. No suction pump, however strong, lifts water from deeper than this. A deeper well needs a pump that pushes from below.
A suction pump draws water from a well whose water surface is \(6\,\text{m}\) below the pump. Take \(P_0 = 10^5\,\text{Pa}\) and \(g = 10\,\text{m/s}^2\). To what absolute pressure must the pump lower the top of the pipe, at the least, in Pa?
Estimate the force of the air on the palm of your hand, in newtons. Take atmospheric pressure as \(10^5\,\text{Pa}\).
The India Mark II hand pump, seen in villages all over the country, often draws water from wells far deeper than \(10\,\text{m}\). It can, because it is not a suction pump. Its cylinder and piston sit down in the well, below the water. Each stroke of the handle lifts the column of water above the piston with a long rod. The water is pushed up from below, not sucked from above.
- Adding \(P_0\) twice, or leaving it out, when a gauge reading is given.A gauge reads \(P - P_0\). Absolute pressure is the reading plus \(P_0\).
- Using the length of a tilted barometer column.The pressure of a column depends on its vertical height alone.
- Comparing two arms of a U-tube at a level where the liquids differ.Compare at the lowest boundary between the liquids, where one liquid joins both arms below.
- Saying that a pump sucks water up.A pump only lowers the pressure above the water. The atmosphere does the pushing, and it can lift water at most about \(10\,\text{m}\).