A rectangular block of wood is 40cm wide, 60cm long and 30cm deep. If its relative density is 0.7kg/m^3 and it floats with its 30cm side vertical, determine: a)the length of the block above the water surface and b)the minimum force required to completely submerge the block
W =0.4 m , L = 0.6 m , H = 0.3 m, ρ1 is the density of wood, ρ2 is the density of the liquid,
relative density is ρ1/ ρ 2 = 0.7 (dimensionless quantity!!!)
m•g = F (buoyant)
m•g = ρ1• V•g = ρ1•W•L•H•g,
F (buoyant) = ρ2• V1•g =
= ρ2•W•L•(H-x)•g.
ρ1•W•L•H•g = ρ2•W•L•(H-x)•g,
ρ1• H = ρ2• (H-x),
x = H•( (ρ1/ ρ2) – 1) =
= 0.3•( (1/0.7) – 1) = 0.129 m.
F + mg=F(buoyant)
F = F(buoyant) - mg = ρ2•W•L•H•g – ρ1•W•L•H•g = W•L•H•g•(ρ2– ρ1)
To determine the length of the block above the water surface, we need to compare the weight of the block to the buoyant force it experiences when floating.
First, let's calculate the weight of the block. The formula for weight is weight = mass × gravity. The density of the block can be calculated using the formula density = mass / volume. We can rearrange this formula to find the mass: mass = density × volume.
Given that the relative density is 0.7 kg/m^3, we can multiply this by the density of water (1000 kg/m^3) to find the density of the block: density_block = relative_density × density_water.
density_block = 0.7 kg/m^3 × 1000 kg/m^3 = 700 kg/m^3
Now, let's calculate the volume of the block. The volume of a rectangular block is given by the formula volume = length × width × depth.
volume = 60 cm × 40 cm × 30 cm
Since all dimensions are in centimeters, we need to convert them to meters to ensure consistent units. 1 meter is equal to 100 centimeters, so:
volume = 60 cm/100(cm/m) × 40 cm/100(cm/m) × 30 cm/100(cm/m) = 0.72 m^3
Now we can calculate the mass of the block: mass_block = density_block × volume.
mass_block = 700 kg/m^3 × 0.72 m^3 = 504 kg
The weight of the block is given by the formula weight_block = mass_block × gravity. Assuming gravity is approximately 9.8 m/s^2:
weight_block = 504 kg × 9.8 m/s^2 = 4945.6 N
When the block floats, it displaces an amount of water equal to its own weight. According to Archimedes' principle, the buoyant force acting on the block is equal to the weight of the water displaced. Therefore, the length of the block above the water surface is equal to the depth of water displaced.
Since the 30 cm side is vertical, the displaced depth of water is also 30 cm.
Now, let's calculate the minimum force required to completely submerge the block. To do this, we need to overcome the weight of the block and provide an additional force to submerge it.
The buoyant force acting on the block when it is completely submerged is equal to the weight of the block. So, the minimum force required to submerge the block is equal to the weight of the block plus the buoyant force.
minimum_force = weight_block + weight_block
minimum_force = 4945.6 N + 4945.6 N = 9891.2 N
Therefore, the length of the block above the water surface is 30 cm, and the minimum force required to completely submerge the block is 9891.2 N.