A cube of edge length 10 cm is just balanced at the interface of two liquids A and B. It is 6 cm in liquid B and 4 cm in liquid A. If A and b has specific gravity 0.6 and 0.4 respectively, then mass of cube is
(1) 240 g
(2) 360 g
(3) 480 g
(4) 540 g

Answer is given option (3)...pls give detailed solutions with explanations...

To solve this problem, we need to apply the principles of buoyancy and the concept of specific gravity. We have a cube that is partially submerged in two different liquids, and we need to find its mass based on the information provided.

Understanding the Problem

We have a cube with an edge length of 10 cm, which means its volume (V) can be calculated as follows:

Volume of the cube (V) = edge length³ = 10 cm × 10 cm × 10 cm = 1000 cm³.

Specific Gravity and Density

The specific gravity (SG) of a substance is the ratio of its density to the density of water. Given that:

• Specific gravity of liquid A (SG_A) = 0.6
• Specific gravity of liquid B (SG_B) = 0.4

We can find the densities (ρ) of the liquids using the formula:

Density (ρ) = Specific Gravity (SG) × Density of water (approximately 1 g/cm³).

• Density of liquid A (ρ_A) = 0.6 g/cm³
• Density of liquid B (ρ_B) = 0.4 g/cm³

Calculating the Buoyant Force

The cube is balanced at the interface, which means the weight of the cube is equal to the total buoyant force acting on it. The buoyant force (F_b) is determined by the volume of the liquid displaced and the density of that liquid.

Since the cube is submerged 6 cm in liquid B and 4 cm in liquid A, we can calculate the volume submerged in each liquid:

• Volume submerged in liquid A (V_A) = area of the base × height submerged in A = (10 cm × 10 cm) × 4 cm = 400 cm³.
• Volume submerged in liquid B (V_B) = area of the base × height submerged in B = (10 cm × 10 cm) × 6 cm = 600 cm³.

Calculating the Buoyant Forces

The buoyant force from liquid A (F_bA) and liquid B (F_bB) can be calculated as follows:

• F_bA = V_A × ρ_A = 400 cm³ × 0.6 g/cm³ = 240 g.
• F_bB = V_B × ρ_B = 600 cm³ × 0.4 g/cm³ = 240 g.

Total Buoyant Force

Now, we can find the total buoyant force:

Total Buoyant Force (F_b) = F_bA + F_bB = 240 g + 240 g = 480 g.

Finding the Mass of the Cube

Since the cube is in equilibrium, the weight of the cube (W_cube) is equal to the total buoyant force:

W_cube = F_b

Therefore, the mass of the cube can be directly determined:

Mass of the cube = 480 g.

Final Answer

The mass of the cube is 480 g, which corresponds to option (3).