A wooden cube with a specific gravity of 0.90 and side length 0.120 m is placed into a
bucket of water and floats upright with its sides in a horizontal or vertical orientation.
What is the mass of the cube, what is the buoyancy force acting on the cube and how
much of the cube projects above the surface?
Let the length above the surface of the water is «x”,
a =0.120 m,
ρ1/ρ2 = 0.9. Since the density of water is ρ2 = 1000 kg/m³,
the density of wood is ρ1 = 900 kg/m³.
m•g = F(buoyancy force),
ρ1•V•g = ρ2•V1 •g
ρ1•a³•g = ρ2•a²•(a-x)•g,
ρ1/ ρ2 = (a-x)/a =0.9.
x = a – 0.9•a = 0.1•a =0.012 m.
The mass of the cube is
m = ρ1•a³ = 900•(0.12)³=1.56 kg.
The buoyancy force is
F = ρ2•a²•(a-x)•g = 1000•(0.12)² •(0.12 -0.012) •9.8 = 15.24 N.
To find the mass of the cube, we can use the concept of specific gravity. Specific gravity is the ratio of the density of a substance to the density of a reference substance, which in this case is water.
Given that the specific gravity of the wooden cube is 0.90, we can say that the density of the wooden cube is 0.90 times that of water. The density of water is approximately 1000 kg/m³.
So, the density of the wooden cube would be: (0.90) * (1000 kg/m³) = 900 kg/m³.
To find the mass of the cube, we need to know its volume. Since it is a cube, its volume can be calculated using the formula: volume = side length³.
Given that the side length of the cube is 0.120 m, the volume of the cube would be: (0.120 m)³ = 0.001728 m³.
Now, to find the mass, we use the formula: mass = density * volume.
Therefore, the mass of the cube would be: 900 kg/m³ * 0.001728 m³ = 1.5552 kg (rounded to four decimal places).
Next, let's determine the buoyancy force acting on the cube. When an object is immersed in a fluid, it experiences an upward force called buoyancy, which is equal to the weight of the fluid displaced by the object.
Since the cube is floating upright with its sides horizontal or vertical, it is displacing its own volume of water. The volume of the cube is 0.001728 m³, so the buoyancy force acting on the cube would be: buoyancy force = density of water * volume of cube * acceleration due to gravity.
The density of water is 1000 kg/m³, the volume of the cube is 0.001728 m³, and the acceleration due to gravity is approximately 9.8 m/s².
Therefore, the buoyancy force acting on the cube would be: 1000 kg/m³ * 0.001728 m³ * 9.8 m/s² = 16.9152 N (rounded to four decimal places).
Finally, to determine how much of the cube projects above the surface, we need to compare the weight of the cube to the buoyancy force.
The weight of the cube can be calculated using the formula: weight = mass * acceleration due to gravity.
With a mass of 1.5552 kg and an acceleration due to gravity of 9.8 m/s², the weight of the cube would be: 1.5552 kg * 9.8 m/s² = 15.2136 N (rounded to four decimal places).
Since the cube floats, the buoyancy force must be equal to the weight of the cube. So, the amount of the cube that projects above the water surface is the difference between the weight of the cube and the buoyancy force.
Therefore, the amount of the cube that projects above the water surface would be: 15.2136 N - 16.9152 N = -1.7016 N (rounded to four decimal places).
Since the result is negative, it means that the entire cube is submerged below the water surface and none of it projects above the surface.