Atmospheric Pressure and Archimedes' Principle

Lessons

Senior 3 · Physics

Atmospheric Pressure and Archimedes' Principle

5 lessons

Lesson 5 of 5

Floating, Sinking and Aerostats

25 min

How does a steel ship float?

Drop a steel bolt into Lake Kivu and it sinks at once. Yet the passenger boats that cross the lake are made of steel and weigh many tonnes. And above the lake, a balloon filled with hot air can rise with people in a basket beneath it. Archimedes' principle from lesson 4 explains all three: what matters is how the weight of a body compares with the upthrust it can get.

Float, hover or sink?

A body in a fluid has two forces on it: its weight $W$ down and the upthrust $U$ up.

If the body is fully immersed and $U > W$ , it rises. As it breaks the surface, less of it is immersed, so $U$ falls, until $U = W$ : it floats, partly immersed.
If $U = W$ when fully immersed, it stays at any depth: it hovers (neutral buoyancy).
If $W > U$ even when fully immersed, it sinks.

For a solid body, $W = \rho_{body} V g$ and, fully immersed, $U = \rho_{fluid} V g$ . The same $V$ and $g$ appear in both, so the test is simply a comparison of densities:

Densities	Result	Example in water
$\rho_{body} < \rho_{fluid}$	Floats	Wood, ice, cooking oil
$\rho_{body} = \rho_{fluid}$	Hovers	A fish or submarine adjusting itself
$\rho_{body} > \rho_{fluid}$	Sinks	Stone, iron, sand

Important

Law of flotation: a floating body displaces its own weight of the fluid in which it floats.

W_{body} = U = \rho_{fluid}\,V_{immersed}\,g \qquad\Rightarrow\qquad \frac{V_{immersed}}{V} = \frac{\rho_{body}}{\rho_{fluid}}

So a body floats with the fraction $\frac{\rho_{body}}{\rho_{fluid}}$ of its volume below the surface.

Worked example 1: a floating block of wood

A block of wood 0.50 m × 0.20 m × 0.10 m has a density of $600\ \mathrm{kg\,m^{-3}}$ . It floats in water. Find (a) its weight; (b) the volume below the surface; (c) the fraction immersed.

V = 0.50 \times 0.20 \times 0.10 = 0.010\ \mathrm{m^3} \qquad \text{(a) } W = 600 \times 0.010 \times 9.8 = 58.8\ \text{N}

\text{(b) } V_{imm} = \frac{W}{\rho_w\,g} = \frac{58.8}{1000 \times 9.8} = 0.0060\ \mathrm{m^3} \qquad \text{(c) } \frac{0.0060}{0.010} = 0.60 = \frac{600}{1000}

Ships and boats

Solid steel is almost eight times as dense as water. A steel hull is shaped into a large hollow bowl, so that even when only its lower part is under water it displaces a very large volume. The boat sinks into the water until the weight of water it pushes aside equals its own weight plus its cargo; the more it carries, the lower it floats. Its average density, steel plus the air inside, is less than that of water.

Fresh water, as in Lake Kivu, is less dense than sea water, so a boat floats a little lower in a lake than it would at sea. Ocean-going ships are marked with load lines showing how deep they may safely float in fresh or salt water, in warm or cold seas.

Worked example 2: loading a boat

An empty boat has a mass of 300 kg. When it floats as low as is safe, it displaces 1.2 m³ of fresh water. What is the greatest mass of passengers and cargo it can carry? (illustrative values)

\text{mass of water displaced} = \rho\,V = 1000 \times 1.2 = 1200\ \text{kg}

By the law of flotation the loaded boat's mass equals the mass of water displaced, so the load is at most

1200 - 300 = 900\ \text{kg}

That is about twelve adults of 75 kg with no luggage. Overloading makes the boat float lower, so a wave can come over the side.

Warning

On any boat, wear a life jacket. A life jacket's foam or air chambers add a large volume at low density, so the upthrust holds your head above water even if you are tired or unconscious. Never board an overloaded boat.

Submarines, fish and hydrometers

A submarine has ballast tanks. To dive, it lets sea water into them, so its weight exceeds the upthrust; to surface, compressed air blows the water out, so its weight falls below the upthrust. With the tanks partly full it can hover.
Many fish have a gas-filled swim bladder; by changing the gas in it they change their volume, and so their upthrust, to hover at different depths.
A hydrometer measures the density of a liquid. It is a sealed glass float weighted at the bottom so that it floats upright. In a denser liquid it needs to displace less liquid to support its weight, so it floats higher. The scale on the stem is read at the liquid surface; its numbers increase downwards.

Hydrometers are used to check the strength of battery acid, the sugar content of fruit juice and beer during brewing, and whether milk has been watered down (a hydrometer for milk is called a lactometer).

Worked example 3: a hydrometer reading

A hydrometer of mass 20 g floats in a liquid with 25 cm³ of its volume below the surface. Find the density of the liquid.

\rho = \frac{m}{V_{imm}} = \frac{0.020}{25 \times 10^{-6}} = 800\ \mathrm{kg\,m^{-3}}

(The mass of liquid displaced equals the hydrometer's mass.) This is about right for paraffin.

Definition

Aerostat: a craft that is lighter than air and is held up by the upthrust of the air, such as a hot-air balloon, a gas balloon or an airship. (An aeroplane, which is held up by the flow of air over its wings, is not an aerostat.)

Aerostats: floating in air

Air is a fluid too, so everything in air receives an upthrust equal to the weight of air it displaces. For most objects this is tiny compared with their weight. A balloon can fly only if its total weight, including the gas inside it, is less than the weight of the air it displaces.

Hot-air balloons heat the air in their envelope with a gas burner. Hot air is less dense than the cooler air around it, so the envelope's contents weigh less than the air they displace. The pilot climbs by burning more gas and descends by letting the air cool or opening a vent at the top.
Gas balloons and airships are filled with helium, which is about seven times less dense than air. Weather services release helium- or hydrogen-filled weather balloons that carry instruments many kilometres up. Hydrogen is even lighter but burns explosively, which is why passenger airships now use helium.

Worked example 4: a hot-air balloon in Kigali

A balloon's envelope holds 2800 m³. In Kigali the outside air at 20 °C has density $0.998\ \mathrm{kg\,m^{-3}}$ ; the hot air inside at 100 °C has density $0.784\ \mathrm{kg\,m^{-3}}$ . What is the greatest total mass of envelope, basket, burner and people the balloon can lift?

U = \rho_{out}\,V\,g = 0.998 \times 2800 \times 9.8 \approx 27\,400\ \text{N}

W_{hot\ air} = \rho_{in}\,V\,g = 0.784 \times 2800 \times 9.8 \approx 21\,500\ \text{N}

\text{spare lift} = 27\,400 - 21\,500 \approx 5900\ \text{N} \quad\Rightarrow\quad m = \frac{5900}{9.8} \approx 600\ \text{kg}

At sea level, where the air is denser (1.204 and 0.946 kg m⁻³ at the same temperatures), the same balloon could lift about 720 kg. A balloon in the thinner air of Kigali must be bigger, or hotter, to carry the same load.

Tip

Compare densities, then use the law of flotation. To decide whether it floats: compare $\rho_{body}$ with $\rho_{fluid}$ . To calculate how it floats: set weight = weight of fluid displaced.

Archimedes' principle: why huge ships float (MooMooMath and Science). Watch for the idea of a hull that displaces a large volume of water, so that the upthrust can match the weight of a steel ship.

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Practice

State the law of flotation.
A block of wood of density $700\ \mathrm{kg\,m^{-3}}$ floats in water. What fraction of its volume is under water? What fraction would be under the surface of a liquid of density $875\ \mathrm{kg\,m^{-3}}$ ?
An empty boat of mass 250 kg carries four passengers of 65 kg each. What volume of fresh water does it displace?
A lactometer of mass 30 g floats in a sample of milk with 29 cm³ immersed. Find the density of the milk. Pure milk has a density of about $1030\ \mathrm{kg\,m^{-3}}$ ; has this sample been watered down?
A ship sails from a freshwater river into the sea. Does it rise or sink a little in the water? Explain.
Explain how a submarine dives and surfaces.
Explain why a hot-air balloon rises, and why it needs a burner to stay up.
A company plans to offer hot-air balloon flights both near sea level and in Kigali, using the same balloon and the same number of passengers. Evaluate whether this will work, using worked example 4.

Show answersHide answers

A floating body displaces its own weight of the fluid in which it floats.
$\frac{700}{1000} = 0.70$ in water; $\frac{700}{875} = 0.80$ in the lighter liquid.
Total mass $250 + 4 \times 65 = 510\ \text{kg}$ , so it displaces 510 kg of water: $V = \frac{510}{1000} = 0.51\ \mathrm{m^3}$ .
$\rho = \frac{0.030}{29 \times 10^{-6}} \approx 1030\ \mathrm{kg\,m^{-3}}$ , the density of pure milk; this sample does not appear watered down (watered milk would be less dense, so the lactometer would sink deeper).
It rises a little. Sea water is denser, so a smaller volume of it weighs as much as the ship; less of the hull needs to be under water.
To dive, it floods its ballast tanks with water, so its weight exceeds the upthrust. To surface, compressed air blows the water out, so its weight becomes less than the upthrust.
The heated air in the envelope is less dense than the surrounding air, so the air displaced weighs more than the hot air plus the balloon: the upthrust exceeds the weight. As the air inside cools it becomes denser and the spare lift disappears, so the burner must keep reheating it.
The same balloon at the same temperatures lifts about 720 kg at sea level but only about 600 kg in Kigali, because the thinner air gives less upthrust. If the load is set for sea level, it would be too heavy in Kigali; the company must carry fewer passengers in Kigali, heat the air more (within safe limits), or use a larger envelope.

Key points

A body floats if $\rho_{body} < \rho_{fluid}$ , hovers if they are equal, and sinks if $\rho_{body} > \rho_{fluid}$ .
Law of flotation: a floating body displaces its own weight of fluid; fraction immersed = $\frac{\rho_{body}}{\rho_{fluid}}$ .
Ships float because their hollow hulls give a low average density; submarines and fish change their weight or volume to dive and rise.
Hydrometers float higher in denser liquids. Aerostats rise when the air they displace weighs more than they do; thinner air at altitude reduces their lift.

1How does a steel ship float?

2Float, hover or sink?

3Ships and boats

4Submarines, fish and hydrometers

5Aerostats: floating in air

How does a steel ship float?

Float, hover or sink?

Ships and boats

Submarines, fish and hydrometers

Aerostats: floating in air