Archimedes' Principle

Archimedes’ principle states that:

An object submerged in a fluid at rest has an upward buoyancy force (upthrust) equal to the weight of the fluid displaced by the object

The object sinks until the weight of the fluid displaced is equal to its own weight
- Therefore, the object floats when the magnitude of the upthrust equals the weight of the object

The magnitude of upthrust can be calculated by:

$F = ρ g V$

Where:
- F = force in newtons (N)
- ρ (Greek letter rho) = density in kilograms per metre cubed (kg m^-3)
- g = gravitational field strength in newtons per kg (N kg^-1)
- V = volume in metres cubed (m³)

Since m = ρV, upthrust is equal to F = mg which is the weight of the fluid displaced by the object
Archimedes’ Principle explains how ships float:

Upthrust on a boat

4-2-6-upthrust-on-a-boat-new

Boats float because they displace an amount of water that is equal to their weight

Worked example

Icebergs typically float with a large volume of ice beneath the water. Ice has a density of 917 kg m^-3 and a volume of V_i.

The density of seawater is 1020 kg m^-3.

What fraction of the iceberg is above the water?

A. 0.10 V_i B. 0.90 V_i C. 0.97 V_i D. 0.20 V_i

Answer:

Step 1: List the known quantities

Density of ice, ρ_i = of 917 kg m^-3
Volume of ice = V_i
Density of seawater, ρ_w = 1020 kg m^-3
Volume of seawater = V_w

Step 2: Consider Archimedes' Principle

According to Archimedes' Principle the force of upthrust is equal to the weight of the seawater displaced by the iceberg

$W_{i} = m_{i} g$

Buoyancy force is the weight of the displaced water

$W_{w} = m_{w} g$

Step 3: Equate the forces of weight and upthrust

Since the iceberg is floating, its weight is exactly equal to the buoyancy force

$W_{i} = W_{w}$

$m_{i} g = m_{w} g$

Step 4: State the density equation and rearrange for mass

$ρ = \frac{m}{V}$

$m = ρ V$

Step 5: Substitute ρV for mass in the mg equivalence

$ρ_{i} V_{i} g = ρ_{w} V_{w} g$

Step 6: Determine the ratio of densities

Cancelling g:

$ρ_{i} V_{i} g = ρ_{w} V_{w} g$

Dividing by $p_{w} V_{i}$

$\frac{ρ_{i}}{ρ_{w}} = \frac{V_{w}}{V_{i}}$

Step 7: Solve for the volume of ice submerged underwater

$V_{w} = \frac{ρ_{i} V_{i}}{ρ_{w}}$

$V_{w} = \frac{917 V_{i}}{1020} = 0.9 V_{i}$

This means that 90% of the iceberg's volume is submerged underwater
The correct answer is A

Examiner Tip

Don't get confused by the two step process to find upthrust.

Step 1: You need the volume of the submerged object, but only because you want to know how much fluid was displaced
Step 2: What you really want to know is the weight of the displaced fluid.

A couple of familiar equations will help;

m = ρV to get mass

then

W = mg to get weight

If you are feeling particularly mathematical, you can combine your equations, so that W = ρVg

Archimedes' Principle (CIE A Level Physics)

Revision Note