doPhysics · Fluids
Pressure in Fluids
What makes a fluid, and the pressure it exerts at a point. Pressure that grows with depth, the atmosphere and the instruments that measure it, Pascal's law and hydraulic machines, and the tilted and curved surfaces of liquids in accelerating and rotating vessels.
Start reading0%done
Why this chapter
Why this chapter exists
Water runs out of a tap on the ground floor with more force than on the top floor. A small push on a garage jack lifts a car. A tanker full of water surges forward when it brakes. All three are one idea. A fluid at rest pushes on everything it touches, and the push changes from point to point by a simple rule. This chapter finds the rule from Newton's second law, applied to a small piece of the fluid. The same rule then works in a lift, on a braking truck and in a spinning cup of tea.
You will learn
By the end of this chapter you will
- Say what makes a fluid, and find the density of a mixture.
- Define pressure at a point, and show that in a fluid at rest it is the same in every direction.
- Derive \(P = P_0 + \rho g h\), and use it for layered liquids, the hydrostatic paradox and the force on a base or a wall.
- Use barometers, manometers and gauges, and tell absolute pressure from gauge pressure.
- State Pascal's law, and find the forces, distances and work in a hydraulic lift, a jack and a brake.
- Find the tilt of a liquid surface in an accelerating vessel, the pressure in a lift, and the levels in an accelerated U-tube.
- Show that the surface of a rotating liquid is a paraboloid, and find its depth at the centre and at the rim.
Before you begin
Before you begin
From the chapters on Newton's laws, you should know how to balance the forces on a body at rest. You should also know \(\Sigma F = ma\) for a body that accelerates. You should know the pseudo force \(-ma_0\) of the chapter on Accelerated Objects, and the centripetal acceleration \(\omega^2 r\) of circular motion. You should be able to integrate a simple function such as \(\int x\,dx\).
Lessons
1
Fluids and Pressure
What makes a liquid or a gas a fluid. Density, relative density and the density of a mixture. Pressure at a point, why a fluid at rest pushes at right angles on every surface, and why that pressure is the same in every direction.
What makes a fluidDensity and relative densityPressure at a point
2
Pressure Grows With Depth
Newton's second law applied to a column of liquid gives \(P = P_0 + \rho gh\). Points at one level share one pressure, so free surfaces are level. The shape of a vessel does not matter, layers add, and the force on a wall grows as the square of the depth.
Pressure variation with depthSame level same pressureHydrostatic paradoxLiquids in layers and force on a base
3
The Atmosphere, Gauges and Barometers
The air is a fluid, and its weight presses on everything at about \(10^5\,\text{Pa}\). Gauge pressure against absolute pressure, the mercury barometer, the open-tube manometer and a U-tube with two liquids. Why a pump that sucks cannot lift water more than about 10 m.
Absolute and gauge pressureThe mercury barometerThe open tube manometerWhy a hand pump lifts water
4
Pascal's Law and Hydraulic Machines
A pressure added at one point of an enclosed liquid reaches every point of it. So a small force on a narrow piston can hold a large force on a wide one. The hydraulic lift, the garage jack and the brakes of a car, and why none of them gives work for nothing.
Pascals lawThe hydraulic liftHydraulic brakes and the jack
5
Liquids in Accelerating Vessels
Newton's second law for a slab of liquid in a vessel that accelerates. The pressure falls along the acceleration, the liquid behaves as if gravity were \(g - a\), and its surface tilts to \(\tan\theta = a/g\). A bucket in a lift, a braking tanker and an accelerated U-tube.
Pressure gradient in an accelerating fluidTilted surface of an accelerating tankBucket in a liftU tube accelerated along its arms
6
Rotating Liquids
A liquid turning with its vessel needs a centripetal force, so its pressure grows away from the axis, \(dP/dr = \rho\omega^2 r\). Its surface becomes a paraboloid, \(y = \omega^2r^2/2g\), whose centre falls as far as its rim rises. Stirred tea, and the centrifuge.
Pressure grows away from the axisThe paraboloid surfaceStirred tea and the centrifuge
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- absolute pressure
- Pressure measured from zero, the pressure of a perfect vacuum.
- atmospheric pressure
- The pressure of the air at a place, caused by the weight of the air above it. At sea level it is about \(1.013 \times 10^5\) Pa, the pressure of 76 cm of mercury.
- barometer
- An instrument that measures atmospheric pressure. The mercury barometer balances the atmosphere against a column of mercury about 76 cm tall.
- centrifuge
- A machine that spins samples fast, so that the effective gravity \(\omega^2 r\) is thousands of times \(g\) and the denser parts move quickly outward.
- density
- Mass per unit volume, \(\rho = m/V\), in kg/m³. Water has a density of 1000 kg/m³, which is 1 g/cm³ or 1 kg per litre.
- effective gravity
- The acceleration \(\vec g - \vec a\) that a liquid in a vessel with acceleration \(\vec a\) behaves as if it had in place of \(\vec g\). Surfaces of equal pressure are at right angles to it.
- fluid
- A substance that cannot resist a shearing force while it is at rest, so that the smallest sideways push makes it flow. Liquids and gases are fluids.
- free surface
- The surface of a liquid that is open to the air or to another fluid above it. Every point of it is at the same pressure.
- gauge pressure
- The amount by which a pressure exceeds atmospheric pressure, \(P - P_0\). Tyre gauges and blood-pressure meters read gauge pressure.
- hydraulic lift
- A machine in which a small force on a narrow piston raises a pressure that pushes a wide piston with a large force. The force is multiplied; the work is not.
- hydrostatic paradox
- The fact that vessels of any shape with equal bases, filled to one depth, have equal forces on their bases, however much liquid each holds.
- manometer
- A U-tube holding a liquid, used to measure the pressure of a gas from the difference between the levels in its two arms.
- paraboloid
- The bowl-shaped surface made by turning a parabola about its axis. A liquid rotating with its vessel takes this shape.
- pascal
- The SI unit of pressure, one newton per square metre, written Pa. Atmospheric pressure is about \(10^5\) Pa.
- Pascal's law
- A change in pressure at one point of an enclosed fluid at rest reaches every point of the fluid, and its walls, undiminished.
- pressure
- The normal force per unit area on a surface, \(P = dF/dA\). In a fluid at rest it is the same in every direction at a point, so it is a scalar.
- pressure gradient
- The rate at which pressure changes with position, and the direction in which it grows fastest. In a fluid at rest it is \(\rho g\), pointing straight down.
- relative density
- The density of a substance divided by the density of water. It has no unit; mercury has a relative density of 13.6.