Question

Two solids A and B float in water . It is observed that A floats with half its volume immersed and B float with 2/3 of its volume immersed . Compare the densities of A and B.

Answer 1

Answer on Question 74132, Physics / Mechanics | Relativity

Question

Two solids A and B float in water. It is observed that A floats with half its volume immersed and B float with 2/3 of its volume immersed. Compare the densities of A and B.

**Solution.** According to the Archimedes' principle the buoyant force $F_{b}$ on an object equals the weight of the water it displaces, $w_{fl}$, that is,

$$F_{b} = w_{fl} = m_{w} g$$

where $m_{w}$ is the displaced mass of water. Let the volumes of the solids A and B be equal, $V_{A} = V_{B} = V$. Buoyant forces for these solids are

$$F_{bA} = m_{wA} g, \quad F_{bB} = m_{wB} g.$$

Here $m_{wA}$ and $m_{wB}$ are the mass of water displaced by these solids

$$m_{wA} = \rho_{w} \cdot \frac{1}{2} V, \quad m_{wB} = \rho_{w} \cdot \frac{2}{3} V$$

where $\rho_{w}$ is the density of water. Then we get for buoyant forces

$$F_{bA} = \frac{1}{2} \rho_{w} g V, \quad F_{bB} = \frac{2}{3} \rho_{w} g V.$$

The masses of solids A and B are

$$m_{A} = \rho_{A} V, \quad m_{B} = \rho_{B} V$$

where $\rho_{A}$ and $\rho_{B}$ are the densities of A and B respectively.

The weights of these bodies are

$$w_{A} = m_{A} g = \rho_{A} g V, \quad w_{B} = m_{B} g = \rho_{B} g V$$

Since these bodies float in water, they are in equilibrium, and weight of the solids is equal to the buoyant force, that is,

$$F_{bA} = w_{A}, \quad F_{bB} = w_{B}$$

or

$$\frac{1}{2} \rho_{w} g V = \rho_{A} g V,$$ $$\frac{2}{3} \rho_{w} g V = \rho_{B} g V$$

Dividing the first equality by the second, we obtain

$$\frac{\rho_{A}}{\rho_{B}} = \frac{1/2}{2/3} = \frac{3}{4}$$

**Answer:** the density ratio is

$$\frac{\rho_{A}}{\rho_{B}} = \frac{3}{4}$$

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