Archimedes' Principle (CIE A Level Physics)

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

$\rho = \frac{m}{V}$

$m = \rho V$

Step 5: Substitute ρV for mass in the mg equivalence 

${\rho }_i{V}_{i}g = {\rho }_w{V}_{w}g$

Step 6: Determine the ratio of densities

• Cancelling g:

${\rho }_i{V}_i\cancel{g} = {\rho }_w{V}_w\cancel{g}$

• Dividing by $p_w{V}_i$

$\frac{{\rho }_i}{{\rho }_w} = \frac{V_w}{V_i}$

Step 7: Solve for the volume of ice submerged underwater

$V_w = \frac{{\rho }_i{V}_i}{{\rho }_w}$

$V_w = \frac{917 V_i}{1020} = 0.9V_i$

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