Course 9.1Lesson 3 of 3

Buoyancy and Archimedes' principle

Archimedes' principle: a body in a fluid is pushed up with a force equal to the weight of fluid it pushes aside, $F_B = \rho_f V g$ . A body sinks if it is denser than the fluid; otherwise it floats with the fraction $\dfrac{\rho_b}{\rho_f}$

about 25 min1 simulation2 practice questions1 Python program4 questions to check yourself

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The push from below

Hold a beach ball under water and you feel it push up. The water presses on every side of a body, but it presses harder lower down, so the push on the bottom is bigger than the push on the top. The difference is the buoyant force.

Archimedes' principle: a body wholly or partly in a fluid is pushed up with a force equal to the weight of the fluid it pushes aside:

\displaystyle F_B = \rho_f V_{\text{under}}\,g

Three tanks of liquid with a block in each. In the first, the block is denser than the liquid, and it sinks. The weight W, downward, is larger than the buoyant force B, upward. In the second, the block is less dense, and it floats with a part above the surface. W and B are equal. In the third, the densities are equal, the block hangs inside the liquid, and W and B are equal. The buoyant force is the weight of the liquid that is pushed aside.

Sink or float

Fully under, a body of density $\rho_b$ has weight $\rho_b Vg$ and buoyant force $\rho_f Vg$ .

$\rho_b > \rho_f$ : it sinks.
$\rho_b < \rho_f$ : it rises, and floats with just enough under to push aside its own weight. Then

\displaystyle \frac{V_{\text{under}}}{V} = \frac{\rho_b}{\rho_f}

You should find that a $900\ \text{kg/m}^3$ block floats with $90\%$ under in water but only $87\%$ in sea water.

A steel ship floats because its hull is hollow: its average density, steel plus the air inside, is less than water's.

Relative density

Weigh a body in air ( $W$ ) and fully in water ( $W'$ ). The loss of weight, $W - W'$ , is the weight of water it pushes aside, of the same volume. So

\displaystyle \text{relative density} = \frac{\rho_b}{\rho_{\text{water}}} = \frac{W}{W - W'}

This is how Archimedes is said to have tested a king's crown.

Watch out. The buoyant force depends on the volume pushed aside and the fluid's density, not on the body's own weight or depth.

A worked example

A crown weighs $9.80\ \text{N}$ in air and $9.25\ \text{N}$ in water. Is it pure gold (relative density $19.3$ )?

Loss of weight. $9.80\ \text{N} - 9.25\ \text{N} = 0.55\ \text{N}$ , the buoyant force.
Relative density. $\dfrac{9.80\ \text{N}}{0.55\ \text{N}} = 17.8$ .
Verdict. Less than $19.3$ : some lighter metal has been mixed in.

Practise with new numbers

Try it in Python

The program runs Archimedes' test on the crown.

Remember these

$F_B = \rho_f V_{\text{under}}\,g$ : the weight of fluid pushed aside.
Floats if $\rho_b < \rho_f$ , with $\dfrac{\rho_b}{\rho_f}$ of its volume under.
Relative density $= \dfrac{W}{W - W'}$ .

Check yourself

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

A $0.40\ \text{m}^3$ log is held fully under water. What is the buoyant force on it?

A block of wood floats with $0.70$ of its volume under water. What is its density?

A steel needle sinks, but a steel ship floats. Why?

A stone weighs $6.0\ \text{N}$ in air and $3.6\ \text{N}$ in water. What is its relative density?

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Problems (4)

A 0.75 kg copper block (density 8900 kg/m³) hangs from a spring balance, fully under water.Easy · High School (11–12) · by StemPy Lab

A raft of wood is 2.5 m × 1.6 m and 0.25 m thick.Medium · High School (11–12) · by StemPy Lab

An old coin weighs 0.315 N in air and 0.285 N when it hangs fully in water.Medium · High School (11–12) · by StemPy Lab

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