Physics › Materials › Fluids: pressure, upthrust and viscosity
Fluids: pressure, upthrust and viscosity
Liquids and gases push on everything inside them, and the push grows with depth. That one fact explains why things float, and a second fact, that real fluids resist being stirred, explains why they fall slowly. Edexcel and CIE examine fluids directly; for everyone else this is where upthrust comes from.
Builds on Density and Hooke's law and Drag and terminal speed.
IN THIS TOPIC
- Use p = ρgh for the pressure due to a column of fluid.
- Explain upthrust as the weight of fluid displaced, and state the condition for floating.
- Use Stokes' law for the viscous drag on a small sphere, and find its terminal velocity.
WHAT YOU PROBABLY THINK
Things float because they are light.
Pressure grows with depth
A fluid pushes on any surface inside it, at right angles to the surface, and the deeper you go the harder the push. The pressure added by a depth h of fluid of density ρ is
which is nothing more than the weight of the fluid column above, divided by the area it stands on. It says pressure depends on depth and density but not on the shape or width of the container: ten metres down feels the same in a lake as in a flooded mine shaft.
WORKED EXAMPLE
Pressure on a diver
Find the extra pressure on a diver 10 m below the surface of fresh water (ρ = 1000 kg m−3).
p = ρgh = 1000 × 9.81 × 10 = 9.8 × 104 Pa.
That is almost exactly one extra atmosphere. Every ten metres of water adds another, which is why depth, not distance, is the thing divers count.
Upthrust: why anything floats
Because pressure grows with depth, the bottom of a submerged object is pushed up harder than its top is pushed down. The imbalance is a net upward force, the upthrust, and its size follows from p = ρgh applied to both faces:
the upthrust equals the weight of fluid displaced by the object. An object floats when it can displace its own weight of fluid before going fully under; it sinks when even fully submerged the displaced fluid weighs less than it does. So floating is not about being light. It is about density: steel sinks as a bar and floats as a ship because the ship's shape displaces far more water.
Viscosity and Stokes' law
Real fluids resist flowing, and the resistance is measured by the viscosity η. Smooth, layered flow is called laminar; above a certain speed it breaks up into chaotic turbulent flow, and the drag laws change completely. For a small sphere of radius r moving slowly enough that the flow around it stays laminar, the viscous drag is given by Stokes' law:
The drag is proportional to speed, so a falling sphere cannot accelerate for ever. It reaches terminal velocity when its weight is balanced by upthrust plus viscous drag.
YOUR TURN
A ball bearing in oil
A steel sphere of radius 2.0 mm and weight 2.6 × 10−3 N falls through oil of viscosity 0.30 Pa s. The upthrust on it is 3.3 × 10−4 N. Find its terminal velocity.
Show the working
At terminal velocity, drag = weight − upthrust = 2.6 × 10−3 − 3.3 × 10−4 = 2.27 × 10−3 N.
Stokes: v = F/6πηr = 2.27 × 10−3 / (6π × 0.30 × 2.0 × 10−3) = 0.20 m s−1.
Slow, steady and measurable, which is exactly why falling-sphere experiments are how viscosity is measured in practice.
Viscosity falls steeply as a liquid warms, which is why cold engines are hard on their oil pumps and why the falling-sphere experiment must record its temperature to mean anything.
THE EXAM BIT
- p = ρgh gives the pressure due to the fluid column alone. If a question wants the total pressure at depth, add atmospheric pressure on top.
- Upthrust arguments score by naming the mechanism: pressure on the lower face exceeds pressure on the upper face, and the resultant equals the weight of fluid displaced.
- Stokes' law holds for small spheres in laminar flow. If a question flags turbulence or a large fast object, the law is the wrong tool and saying so earns the mark.
- Terminal velocity working starts from the force balance: weight = upthrust + drag. Write it before substituting anything.
- Fluids are examined directly by Edexcel and CIE, and density and pressure by OCR A. AQA stops at the qualitative drag ideas in the mechanics unit.
CHECK YOURSELF
A wooden block of density 600 kg m−3 floats in water of density 1000 kg m−3. What fraction of the block is submerged?
Show a hint
Floating means the displaced water weighs exactly what the block weighs.
Show the answer
Floating: weight of block = weight of displaced water, so ρblockVwholeg = ρwaterVsubg.
Vsub/Vwhole = 600/1000 = 0.6. Three fifths of the block sits below the waterline, whatever its size or shape.
Pressure in a fluid is depth times density times g.
Upthrust is the weight of fluid displaced; drag on a slow sphere is Stokes' law.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Use p = ρgh for the pressure due to a column of fluid.
- Explain upthrust as the weight of fluid displaced, and state the condition for floating.
- Use Stokes' law for the viscous drag on a small sphere, and find its terminal velocity.
Open the full revision checklist to track your progress across the whole unit.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.