A uniform wooden cube of edge length 0.20\text{ m} floats in water of density 1000\text{ k… — JAMB UTME

A uniform wooden cube of edge length $0.20\text{ m}$ floats in water of density $1000\text{ kg/m}^3$ with $0.15\text{ m}$ of its vertical height submerged. Taking acceleration due to gravity $g = 10\text{ m/s}^2$ , what minimum mass, in kilograms, must be placed on the top surface of the cube so that its upper face becomes just flush with the water surface?

Answer: 2 kg

Answer

The minimum mass required to submerge the cube completely flush with the water surface is

2.0\text{ kg}

By the Law of Flotation, a floating body displaces its own weight of fluid. Initially, the cube displaces a volume of

0.20\text{ m} \times 0.20\text{ m} \times 0.15\text{ m} = 0.006\text{ m}^3

of water, corresponding to an upthrust of

60\text{ N}

(or mass of

6.0\text{ kg}

). When completely submerged, the total volume displaced is

0.20^3 = 0.008\text{ m}^3

, providing a total upthrust of

80\text{ N}

(or mass equivalent of

8.0\text{ kg}

). The additional mass required on top is therefore the difference:

8.0\text{ kg} - 6.0\text{ kg} = 2.0\text{ kg}

Step-by-Step Solution

Calculate the cross-sectional area of the cube

A = (0.20\text{ m})^2 = 0.04\text{ m}^2

The base area is needed to find the volume of the block submerged and unsubmerged.

Calculate the volume of the cube above the water surface

V_{\text{above}} = 0.04\text{ m}^2 \times (0.20\text{ m} - 0.15\text{ m}) = 0.002\text{ m}^3

To push the cube level with the surface, the additional weight added on top must balance the extra upthrust created by submerging this remaining volume.

Calculate the additional mass required

m = \rho_{\text{water}} \times V_{\text{above}} = 1000\text{ kg/m}^3 \times 0.002\text{ m}^3 = 2.0\text{ kg}

By Archimedes' principle, the additional downward mass must equal the mass of the extra water displaced when fully submerged.

Key Concept

Archimedes' Principle and Law of Flotation

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic