A 40mm x 60mm x 3.5m long wooden block having a specific gravity of 0.60 is made to float in water. If a stone (s.g.=3.25) should be attached at the bottom to make the block to float 300mm exposed to atmosphere. Calculate the amount of stone in kg.

Answer 1

First, we need to find the volume of the wooden block.

Volume (V) = Length × Width × Height
V = 0.06m × 0.04m × 3.5m = 0.0084 m³.

Next, we'll find the mass of the wooden block using its specific gravity (sg). We know that a specific gravity of 1.0 means that an object is neutrally buoyant in water, and the weight of the water displaced by the object is equal to the weight of the object itself. So, we can find the mass of the wooden block using its specific gravity:

Mass (m_wood) = V × sg × ρ_water
m_wood = 0.0084 m³ × 0.60 × 1000 kg/m³ = 5.04 kg,

where ρ_water is the density of water and is equal to 1000 kg/m³.

Now, we need to find the volume of the wooden block that is submerged in water when the block is floating with 300mm exposed to the atmosphere. The total length of the block is 3.5m, and the exposed length is 0.3m, so the submerged length is:

Submerged length = 3.5m - 0.3m = 3.2m.

Since the width and height of the block are the same for the submerged and total volume, we can find the volume of the submerged portion by multiplying the original volume by the ratio of the submerged length to the total length:

V_submerged = V × (3.2m / 3.5m) = 0.0084 m³ × (3.2m / 3.5m) = 0.00768 m³.

Now we'll find the weight of the water displaced by the submerged portion of the block:

Weight_water = V_submerged × ρ_water × g
Weight_water = 0.00768 m³ × 1000 kg/m³ × 9.81 m/s² = 75.24 N,

where g is the acceleration due to gravity (about 9.81 m/s²).

To find the mass of the stone, we'll first find the buoyant force acting on the stone while it's attached to the wooden block:

Buoyant force = Weight_water - Weight_wood
Buoyant force = 75.24 N - (5.04 kg × 9.81 m/s²) = 75.24 N - 49.49 N = 25.75 N.

Now we'll find the volume of the stone that is submerged, using its specific gravity:

Volume_stone = Volume_water - Volume_submerged
Volume_stone = 0.00768 m³ × (1 / 0.60 - 1) = 0.00768 m³ × (1.67 - 1) = 0.005128 m³.

Next, we'll find the mass of the submerged portion of the stone using its specific gravity:

Mass_submerged = Volume_stone × 3.25 × ρ_water
Mass_submerged = 0.005128 m³ × 3.25 × 1000 kg/m³ = 16.67 kg.

Finally, we'll find the total mass of the stone using the buoyant force:

Mass_stone = Buoyant force / g
Mass_stone = 25.75 N / 9.81 m/s² = 2.63 kg.

Therefore, the amount of stone in kilograms is 2.63 kg.