Buoyancy, floating and Archimedes' principle

Things feel lighter in water

Lift a friend in a swimming pool: it is easy. Push an empty bottle under water: it pushes back up. A liquid pushes up on anything in it. This upward force is the buoyant force, or upthrust.

It comes from depth: the liquid pushes harder on the bottom of the object, which is deeper, than on its top.

Archimedes' principle

An object wholly or partly in a liquid feels an upward force equal to the weight of the liquid it pushes aside.

Float or sink?

• If the object is less dense than the liquid, it floats, with just enough of it under the surface to push aside its own weight of liquid.
• If it is denser, the buoyant force is less than its weight, and it sinks.

The part under the surface is $\dfrac{\rho_\text{object}}{\rho_\text{liquid}}$.

You should find that the block sinks lower as its density approaches the liquid's, and sinks altogether when it is denser. In salt water, a block floats a little higher.

Why does an iron ship float?

Iron is much denser than water, but a ship is mostly air inside. Its average density, iron and air together, is less than water's. Its wide hull pushes aside a huge weight of water.

Watch out. The buoyant force depends on the volume of liquid pushed aside, not on what the object is made of.

A worked example

Do pine wood ($550\ \text{kg/m}^3$), ice ($920\ \text{kg/m}^3$), rubber ($1200\ \text{kg/m}^3$) and iron ($7900\ \text{kg/m}^3$) float in water? How much of each floater is under the surface?

1. Pine. Floats: $\dfrac{550\ \text{kg/m}^3}{1000\ \text{kg/m}^3} = 0.55$, so $55\%$ under.
2. Ice. Floats: $92\%$ under, which is why most of an iceberg is hidden.
3. Rubber and iron. Denser than water: they sink.

Practise with new numbers

Try it in Python

The program sorts the four materials.

Remember these

• A liquid pushes up with a force equal to the weight of liquid pushed aside (Archimedes).
• Less dense than the liquid: floats; denser: sinks.
• Part under the surface $= \dfrac{\rho_\text{object}}{\rho_\text{liquid}}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A stone pushes aside $0.3\ \text{kg}$ of water. What is the buoyant force on it?

2. Why does a ship float higher in sea water than in a river?

3. A block floats with $70\%$ of it under water. What is its density?

4. Why does a steel needle sink while a steel ship floats?