Calculation of the Depth of Immersion of a Rectangular Block in Oil

Given data:
- Length of rectangular block (L) = 6m
- Width of rectangular block (B) = 4m
- Height of rectangular block (H) = 2m
- Specific gravity of oil (S) = 0.90
- Metacentric height (GM) = 1.4m

We need to find the depth of immersion (d) of the rectangular block in oil.

1. Determine the weight of the rectangular block
The weight of the rectangular block can be calculated using the formula:
Weight = Volume x Density x Gravity
- Volume (V) = L x B x H = 6m x 4m x 2m = 48m³
- Density (ρ) = 1,000 kg/m³ (density of water)
- Gravity (g) = 9.81 m/s²
Weight = 48m³ x 1,000 kg/m³ x 9.81 m/s² = 470,880 N

2. Determine the buoyant force acting on the block
The buoyant force acting on the block can be calculated using the formula:
Buoyant force = Volume displaced x Density of fluid x Gravity
- Volume displaced (Vd) = L x B x d
- Density of fluid (S) = 0.90 x 1,000 kg/m³ = 900 kg/m³
- Gravity (g) = 9.81 m/s²
Buoyant force = (L x B x d) x 900 kg/m³ x 9.81 m/s²

3. Determine the position of the center of gravity of the block
The center of gravity of the block can be assumed to be at the geometric center, which is at half of the height, i.e. at H/2 = 1m.

4. Determine the position of the metacenter
The metacenter is located at the intersection of the vertical line passing through the center of buoyancy and the line passing through the center of gravity of the block when it is tilted slightly. In this case, the block is rectangular and symmetrical, so the center of buoyancy coincides with the center of the block, which is at (L/2, B/2, d/2). Therefore, the metacenter is also at (L/2, B/2, GM+d/2).

5. Determine the righting moment
The righting moment is the product of the buoyant force and the distance between the metacenter and the center of gravity, which is (GM + d/2 - 1m).
Righting moment = Buoyant force x (GM + d/2 - 1m)

6. Find the equilibrium condition
For stable equilibrium, the righting moment must be greater than or equal to the overturning moment. The overturning moment is the product of the weight of the block and half of its height, which is 235,440 Nm.
Righting moment ≥ Overturning moment
Buoyant force x (GM + d/2 - 1m) ≥ 235,440 Nm

7. Solve for the depth of immersion
Substituting the values and solving for d, we get:
d ≥ 0.63m