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.
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.
To determine the weight of the stone required to make the wooden block float with 300mm (0.3m) exposed above the water level, we need to consider the buoyant force acting on the block.
1. Calculate the volume of the wooden block:
- Volume = Length x Width x Height
- Volume = 3.5m x 0.04m x 0.06m
- Volume = 0.0084 m^3
2. Calculate the weight of the wooden block:
- Weight = Specific Gravity x Density of Water x Volume of block x g (acceleration due to gravity)
- Weight = 0.60 x 1000 kg/m^3 x 0.0084 m^3 x 9.8 m/s^2
- Weight = 49.392 kg
3. Determine the buoyant force acting on the block:
- Buoyant Force = Weight of Displaced Water
- The volume of water displaced by the block is equal to its volume, which is 0.0084 m^3.
- Buoyant Force = Density of Water x Volume of block x g
- Buoyant Force = 1000 kg/m^3 x 0.0084 m^3 x 9.8 m/s^2
- Buoyant Force = 82.32 N
4. We want the block to float with 300mm of its height above the water level, which means the effective weight of the block should be equal to the buoyant force:
- Effective Weight = Weight of block - Weight of stone
- Effective Weight = 49.392 kg - Weight of stone
5. We know that the stone has a specific gravity of 3.25, which means its density is 3.25 times greater than the density of water:
- Density of Stone = Specific Gravity of Stone x Density of Water
- Density of Stone = 3.25 x 1000 kg/m^3
- Density of Stone = 3250 kg/m^3
6. Calculate the weight of the stone:
- Weight of Stone = Density of Stone x Volume of stone x g
- Volume of stone = Volume of block (since the stone occupies the same space as the part of the block submerged in water)
- Weight of Stone = 3250 kg/m^3 x 0.0084 m^3 x 9.8 m/s^2
- Weight of Stone = 27.0644 kg
7. Now, we can solve for the weight of the stone, which will give us the amount of stone needed:
- Effective Weight = Weight of block - Weight of stone
- 82.32 N = 49.392 kg - Weight of stone
- Weight of stone = 49.392 kg - 82.32 N
- Weight of the stone = 49.392 kg - 8.406244 kg (converting the buoyant force from newtons to kg using g = 9.8 m/s^2)
- Weight of the stone ≈ 41.986 kg
Therefore, the amount of stone required to make the block float with 300mm exposed above the water level is approximately 41.986 kg.
To calculate the amount of stone needed to make the wooden block float with 300mm exposed to the atmosphere, we can start by determining the volume of the wooden block.
1. Calculate the volume of the wooden block:
Volume = Length x Width x Height
Volume = 3.5m x (40mm/1000m) x (60mm/1000m)
Volume = 0.0084 m^3
2. Calculate the volume of water displaced by the wooden block to make it float:
Volume displaced = Volume of the wooden block x Specific gravity of water / Specific gravity of the wooden block
Volume displaced = 0.0084m^3 x (1/1) / (0.60/1)
Volume displaced = 0.0084m^3 x 1.67
Volume displaced = 0.0140m^3
3. Calculate the volume of the stone needed to displace the remaining volume:
Volume of stone = Total volume displaced - Volume of the wooden block
Volume of stone = 0.0140m^3 - 0.0084m^3
Volume of stone = 0.0056m^3
4. Calculate the mass of the stone using the volume and specific gravity:
Mass of stone = Volume of stone x Specific gravity of the stone x Density of water
Mass of stone = 0.0056m^3 x 3.25 x 1000kg/m^3
Mass of stone = 18.2 kg
Therefore, the amount of stone needed to make the wooden block float 300mm exposed to the atmosphere is approximately 18.2 kg.