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Archimedes' principle

Air and fluids · level 5, lesson 4 of 6

A question to start

A stone weighs 30 N in air but only 18 N when it hangs in water. Where did the other 12 N go?

The idea

The pressure under a liquid grows with depth, so the liquid pushes harder on the bottom of an object than on its top. The difference is an upward force: the upthrust.

Archimedes' principle: the upthrust on an object in a fluid equals the weight of the fluid it pushes aside (displaces):

\[U = \rho_\text{fluid}\, V g,\]

where \(V\) is the volume of the object under the surface.

spring balancewater pushed aside:its weight = upthrust

The object's weight is \(mg\). If the upthrust is less, it sinks; if the upthrust, with the object fully under, is more, it rises and floats.

Worked examples

1. The stone. It loses 12 N, so it pushes aside 12 N of water: \(V = 12/(1000 \times 9.81) = 0.00122\) m³, 1.22 litres.

2. The same stone on Mars. Weight \(30 \times 3.71/9.81 = 11.3\) N. Upthrust \(1000 \times 0.00122 \times 3.71 = 4.5\) N. In water it reads \(11.3 - 4.5 = 6.8\) N. Both shrink in the same proportion.

3. In air. The stone also displaces 1.22 litres of air: on Earth that is \(1.217 \times 0.00122 \times 9.81 = 0.015\) N of upthrust. Tiny, but real.

Try it

  1. State Archimedes' principle.
  2. A block of volume 0.002 m³ is fully under water on Earth. Find the upthrust.
  3. A 50 N object reads 35 N when fully under water. What is its volume?
  4. Find the upthrust on the block in question 2 on Mars.
Answers
  1. The upthrust on an object in a fluid equals the weight of the fluid it displaces.
  2. \(1000 \times 0.002 \times 9.81 = 19.6\) N.
  3. Upthrust 15 N, so \(V = 15/(1000 \times 9.81) = 0.00153\) m³.
  4. \(1000 \times 0.002 \times 3.71 = 7.4\) N.

Common mistake

"Upthrust depends on the object's weight." It depends only on the volume under the surface and the fluid's density. A lead block and a plastic block of the same size, both fully under water, feel the same upthrust.

Where it leads