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Archimedes' Principle

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Why this chapter
Why this chapter exists

A steel ship of fifty thousand tonnes floats, while a steel nail sinks. A bucket is light under the water of a well and heavy above it. A boat turns back upright after a wave, and a boat full of standing people may not. All of these follow from one fact. The pressure in a fluid grows with depth, so the fluid pushes harder on the bottom of a body than on its top. This chapter turns that fact into Archimedes' principle and uses it to float, weigh and measure. It ends with what keeps a floating body the right way up.

You will learn
By the end of this chapter you will
  1. Derive the buoyant force from the pressure difference across a body, and state Archimedes' principle with the replacement argument.
  2. Find the apparent weight of a body in a fluid, and decide whether a body sinks or rises from the densities.
  3. Use the law of flotation to find how deep a body floats and what fraction of it is under. Explain why a ship rides higher in the sea.
  4. Explain why a hollow steel ship floats through its average density.
  5. Find the density of a solid and of a liquid by weighing in air and in water, and explain the hydrometer and the lactometer.
  6. Show that a scale under a vessel gains exactly the weight a body loses in it.
  7. Settle the level of water when ice melts or a load is moved on or off a floating body.
  8. Find the buoyant force in an accelerating vessel and on a balloon in air.
  9. Use the centre of buoyancy and the metacentre to decide whether a floating body is stable, and state the condition for a submerged body.
Before you begin
Before you begin

From the chapter on Pressure in Fluids you should know density, relative density and the pressure at a depth, \(P = P_0 + \rho g h\). You should also know \(P = P_0 + \rho (g + a) h\) in an accelerating lift. From Dynamics you should know how the forces on a body at rest balance. You should know Newton's third law and the pseudo force \(-ma_0\) in an accelerating frame. From Rotational Mechanics you should know the centre of gravity and the couple.

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Lessons

1
The Buoyant Force
A fluid pushes harder on the lower face of a body than on its upper face, so it pushes the body up. The buoyant force equals the weight of the fluid displaced, which is Archimedes' principle, and a body weighed in a fluid reads less than its weight.
Buoyancy from the pressure differenceArchimedes principleApparent weight
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2
Floating: the Law of Flotation
A floating body sinks until it displaces its own weight of liquid. The fraction under the surface is the ratio of the densities, which is why an iceberg hides most of its bulk and a ship rides higher in the sea. A steel ship floats because its average density is small.
Law of flotationFraction submergedWhy a steel ship floats
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3
Weighing With Water
Two readings of a balance, in air and in water, give the density of a solid. A third reading in another liquid gives that liquid's density. The hydrometer and the dairy lactometer read density from how deep they float, and the weight a body loses in water appears on a scale under the vessel.
Density by loss of weightDensity of a liquid by weighingThe hydrometer and lactometerWhere the lost weight goes
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4
Buoyancy in Harder Cases
Ice melting in a full glass, a load moved on and off a floating block, a body floating in an accelerating lift, and balloons in air. Each case is settled by one question, how much fluid is displaced, and by the effective gravity \(g + a\).
Ice melting in waterLoad dropped from a floating blockBuoyancy in an accelerating vesselBalloons and buoyancy in air
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5
Stability of Floating Bodies
A tilted boat is acted on by its weight through G and the buoyant force through the centre of buoyancy B, which moves to the low side. The metacentre M decides whether the couple rights the boat, and a submerged body is stable only with B above G.
Centre of buoyancyMetacentre and the righting coupleStability of a submerged body
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

apparent weight
The reading of a balance that holds a body at rest in a fluid, equal to its weight minus the buoyant force, \(W - \rho_f V g\).
Archimedes' principle
A body wholly or partly immersed in a fluid is pushed up by a force equal to the weight of the fluid it displaces.
average density
The total mass of a body divided by the whole volume it encloses, air spaces included. A hollow body floats when its average density is less than the fluid's.
buoyant force
The resultant of all the pushes of a fluid on the surface of a body in it. It acts upward and equals the weight of the fluid displaced, \(F_B = \rho_f V_{sub} g\). Also called buoyancy or upthrust.
centre of buoyancy
The point through which the buoyant force acts: the centre of gravity of the displaced fluid. It moves when the shape of the displaced fluid changes.
couple
Two equal and opposite forces that do not act along one line. A couple turns a body without moving it along.
displaced fluid
The fluid that a body pushes out of the space it occupies below the surface. Its volume is the volume of the body under the surface.
effective gravity
The gravity felt inside a vessel accelerating upward at \(a\), equal to \(g + a\). Pressure differences, weights and buoyant forces all carry it in place of \(g\).
flotation
The state of a body floating at rest, with the buoyant force on its submerged part equal to its weight.
hydrometer
A weighted float with a thin graduated stem that floats upright in a liquid. The mark at the surface gives the liquid's density, the smaller values higher on the stem.
lactometer
A hydrometer made for milk. Watered milk is less dense than pure milk, so the lactometer sinks deeper in it.
law of flotation
A body floating at rest displaces a weight of fluid equal to its own weight.
load line
A set of marks painted on a ship's hull. They show how deep the loaded ship may safely lie in fresh water and in sea water. Also called the Plimsoll line.
metacentre
For a tilted floating body, the point where the vertical line through the centre of buoyancy meets the body's centre line. The body is stable when the metacentre lies above its centre of gravity.
metacentric height
The distance \(GM\) from the centre of gravity up to the metacentre. For small tilts the righting couple is \(W \cdot GM \sin\theta\).
relative density
The density of a substance divided by the density of water. It has no unit, and for a solid it equals its weight in air divided by its loss of weight in water.