Fill a plastic water bottle and make three small holes around its side with a hot pin, all at the same height. Water jets out of every hole, sideways. Each jet leaves at right angles to the wall, whichever way the wall faces. The water in the bottle is at rest, yet it pushes outward on the side, not only down on the base.
Solids, liquids and gases
The molecules of a solid are held in fixed places by strong forces between them. So a solid keeps both its shape and its volume. In a liquid the forces are weaker. The molecules stay close together, but they can slide past one another. So a liquid keeps its volume but takes the shape of its container. In a gas the molecules are far apart and almost free. A gas spreads out to fill whatever holds it.
What makes a fluid
Hold the bottom of a rubber block and push its top face sideways. The block bends a little and then pushes back, and it stays bent while you push. A push along a surface like this is a shearing force. Now push sideways on the top layer of water in a tray. The layer slides over the one below, and it keeps sliding while you push. Water cannot hold a shearing force at all.
A fluid is a substance that cannot resist a shearing force while it is at rest. However small the sideways push, its layers slide and it flows.
Liquids and gases both flow, so both are fluids. Honey flows slowly and water quickly, but both flow under the smallest push. The difference is in how fast they flow, and the chapter on Viscosity studies it. At rest, honey and water obey the same rules.
The line between solid and fluid depends on how long you wait. Pitch, a tar used on roads, shatters under a hammer blow. Yet a funnel of pitch set up in 1927 at the University of Queensland still drips. Fewer than ten drops have fallen since then.
A fluid at rest pushes at right angles
A fluid at rest cannot push along a surface, since any such force would make it flow. So the force of a fluid at rest on any surface has no part along that surface. The force is always at right angles, or normal, to the surface. It is the same for the wall of a bottle, the face of a dam and the skin of a diver.
A thick layer of honey rests on a plate. You tilt the plate slightly, so the honey's weight has a small part along the plate. What happens?
Density
A fluid has no shape of its own, so a mass alone says little about it. A better measure is the mass in each unit of volume. This is the density \(\rho\) (the Greek letter rho) of the substance. For a uniform sample of mass \(m\) and volume \(V\), \(\rho = m/V\). The SI unit is kg/m³.
Water has a density of \(1000\,\text{kg/m}^3\). The same number reads \(1\,\text{g/cm}^3\), or \(1\,\text{kg}\) per litre, which is easy to remember. Kerosene is about \(800\,\text{kg/m}^3\), sea water and milk about \(1030\), aluminium \(2700\), iron about \(7800\) and mercury \(13\,600\). Air at room temperature is only about \(1.2\,\text{kg/m}^3\).
A density is often given by comparing it with water. The relative density of a substance is its density divided by the density of water. It is a ratio of two densities, so it has no unit. Aluminium has a relative density of \(2.7\), which means its density is \(2.7 \times 1000 = 2700\,\text{kg/m}^3\). Mercury has a relative density of \(13.6\).
The relative density of a substance is \(\rho/\rho_{\text{water}}\), its density divided by the density of water. It is a pure number, and it is also called specific gravity.
The density of a mixture
When two liquids are mixed, the mixture has one density. It is the total mass divided by the total volume. We assume here that the volumes simply add, which is nearly true for most pairs of liquids. For equal volumes the density of the mixture is the plain average. For equal masses it is less than the average, because the lighter liquid takes up more of the room.
Two liquids have densities \(800\,\text{kg/m}^3\) and \(1200\,\text{kg/m}^3\). Assume the volumes add when they mix. Find the density of the mixture (a) of \(1\,\text{L}\) of each, and (b) of \(1\,\text{kg}\) of each.
(a) The masses are \(0.8\,\text{kg}\) and \(1.2\,\text{kg}\), \(2\,\text{kg}\) in \(2\,\text{L}\). So \(\rho = 1\,\text{kg/L} = 1000\,\text{kg/m}^3\), the plain average.
(b) The volumes are \(1/800 = 0.00125\,\text{m}^3\) and \(1/1200 \approx 0.000833\,\text{m}^3\), together \(0.002083\,\text{m}^3\). So \(\rho = 2/0.002083 = 960\,\text{kg/m}^3\).
In (b) the lighter liquid fills three fifths of the volume, so it pulls the density below the average. In general, equal masses give \(2\rho_1\rho_2/(\rho_1 + \rho_2)\).
A milkman adds \(1\,\text{L}\) of water to \(2\,\text{L}\) of milk of density \(1030\,\text{kg/m}^3\). Assume the volumes add. What is the density of the mixture, in kg/m³?
A classroom is \(8\,\text{m}\) long, \(6\,\text{m}\) wide and \(3.5\,\text{m}\) high. Air has a density of about \(1.2\,\text{kg/m}^3\). Estimate the mass of the air in the room, in kg.
Pressure
A fluid pushes on every surface it touches, and the push is spread over the whole surface. So the useful quantity is the push on each unit of area. The pressure \(P\) on a flat surface is the normal force \(F\) on it divided by its area \(A\), \(P = F/A\). Its SI unit is the newton per square metre, called the pascal (Pa).
Pressure matters for solids too. A student of mass \(60\,\text{kg}\) stands on two feet, each touching the floor over \(150\,\text{cm}^2\). The weight \(600\,\text{N}\) is spread over \(0.03\,\text{m}^2\), so the pressure on the floor is \(2 \times 10^4\,\text{Pa}\). On one foot the same weight presses on half the area, and the pressure doubles to \(4 \times 10^4\,\text{Pa}\).
Take a small flat area \(dA\) around a point, and let \(dF\) be the normal force on one side of it. The pressure at the point is \(P = dF/dA\), the limit as the area shrinks to the point.
Where the push comes from
The molecules of a fluid are always moving, in every direction and at high speeds. They strike every surface in the fluid, the walls of the container and the faces of anything placed in it. Each collision gives a tiny push at right angles to the surface. Billions of collisions every second add up to a steady force on each unit of area. That force per area is the pressure.
A gauge that turns
To see how pressure behaves, imagine a small gauge. It is a short cylinder with a vacuum inside, closed by a light piston held out by a spring. The fluid pushes the piston in until the spring pushes back equally. So the compression of the spring measures the force on the piston, and therefore the pressure at that point.
Watch the piston in Fig 1.1 while the gauge turns. With the face up, the water pushes the piston down. With the face sideways or down, the water pushes it sideways or up, by exactly as much. The reading depends on where the gauge is, not on which way its face points. Only when the gauge goes deeper does the piston move further in.
The same in every direction
Why should the pressure not depend on direction? Take a tiny wedge of fluid at rest, as in Fig 1.2. Its three faces are a vertical face of height \(dy\), a horizontal base of width \(dx\) and a slanted face of length \(ds\). The wedge is a length \(l\) deep into the page. The fluid around it pushes on each face at right angles, with pressures \(P_1\), \(P_2\) and \(P_3\).
\[ F_1 = P_1\,l\,dy, \quad F_2 = P_2\,l\,dx, \quad F_3 = P_3\,l\,ds \]
Why. each force is the pressure on a face times the area of that face.
\[ F_1 = F_3\sin\theta \]
Why. horizontally only \(F_1\) and the horizontal part of \(F_3\) act, and the wedge is at rest.
\[ P_1\,l\,ds\sin\theta = P_3\,l\,ds\sin\theta \;\Rightarrow\; P_1 = P_3 \]
Why. put \(dy = ds\sin\theta\) and cancel \(l\,ds\sin\theta\).
\[ F_2 = F_3\cos\theta + \tfrac12 \rho g\,l\,dx\,dy \]
Why. vertically, \(F_2\) holds up the vertical part of \(F_3\) and the weight of the wedge.
\[ P_2 = P_3 + \tfrac12 \rho g\,dy \;\to\; P_2 = P_3 \]
Read it. divide by \(l\,dx = l\,ds\cos\theta\). As the wedge shrinks to a point, \(dy \to 0\) and the weight term vanishes.
At any point of a fluid at rest, the pressure is the same on a small surface facing any direction. So pressure is a scalar. The force it produces on a surface has a direction: at right angles to that surface.
Return to the bottle of the opening. At the height of the holes, the water presses on the wall with the same pressure as it presses on anything else at that height. The wall holds that push until a hole opens. Then nothing holds the water there, and it jets out at right angles to the wall, the direction of the push.
The piston of the gauge has an area of \(2\,\text{cm}^2\). At the depth where the pressure is \(1.5 \times 10^5\,\text{Pa}\), what force does the water exert on it, in newtons?
A student writes that pressure is a vector, because a fluid pushes down on the base of a beaker and sideways on its wall. What is wrong with this?
Units of pressure
The pascal is a small unit. A sheet of paper lying on a table presses on it with only about \(1\,\text{Pa}\). So larger units are common. One bar is \(10^5\,\text{Pa}\). One atmosphere (atm) is \(1.013 \times 10^5\,\text{Pa}\), the average pressure of the air at sea level. It equals the pressure of a column of mercury \(76\,\text{cm}\) tall, so pressures are also quoted in centimetres or millimetres of mercury.
One millimetre of mercury is about \(133\,\text{Pa}\). A doctor who reads your blood pressure as \(120\) over \(80\) means millimetres of mercury above the air's pressure. So \(120\,\text{mm}\) of mercury is about \(1.6 \times 10^4\,\text{Pa}\). Tyre gauges at petrol pumps often read in psi, pounds per square inch; one psi is about \(6.9 \times 10^3\,\text{Pa}\).
Arrange these pressures from the smallest to the largest.
A school bag of \(60\,\text{N}\) hung from thin strings would cut into the shoulders. Wide padded straps carry the same weight over a larger area, so the pressure on the skin is much smaller. The same idea puts wide tyres on a tractor that must cross soft soil, and a sharp edge on a kitchen knife that must cut.
- Thinking that a fluid at rest can push along a surface.A fluid at rest cannot hold a shearing force. Its force on any surface is at right angles to that surface.
- Treating pressure as a vector that points down.Pressure at a point is the same in every direction. Only the force on a surface has a direction, normal to the surface.
- Averaging the densities of a mixture of equal masses.Always divide the total mass by the total volume. Only equal volumes give the plain average.
- Forgetting to convert cm² to m² in \(P = F/A\).\(1\,\text{cm}^2 = 10^{-4}\,\text{m}^2\). A force of \(1\,\text{N}\) on \(1\,\text{cm}^2\) is \(10^4\,\text{Pa}\).