To determine how much of the height of the cylinder is in the oil, we need to compare the densities of the cylinder, water, and oil.
The density of an object is defined as the mass of the object divided by its volume. For the cylinder, we have its mass and dimensions. The volume of a solid cylinder is given by the formula V = πr²h, where r is the radius and h is the height.
Given: Radius of the cylinder, r = 17.5 cm = 0.175 m Height of the cylinder, h = 16.5 cm = 0.165 m Mass of the cylinder, m = 13.49 kg
Let's calculate the volume of the cylinder: V_cylinder = πr²h = π(0.175 m)²(0.165 m) ≈ 0.1605 m³
Now, let's determine the density of the cylinder: Density_cylinder = mass_cylinder / volume_cylinder = 13.49 kg / 0.1605 m³ ≈ 83.99 kg/m³
The density of water is approximately 1000 kg/m³, and the density of oil varies depending on the type of oil used. Let's assume the density of the oil is ρ_oil kg/m³.
Since the cylinder is floating, the upward buoyant force exerted on it by the water and oil combined must equal the downward force due to gravity acting on the cylinder.
The buoyant force is given by Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object.
The weight of the cylinder is given by W_cylinder = mass_cylinder * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
The weight of the fluid displaced is given by W_fluid = (volume_submerged in fluid) * (density_fluid) * g.
Since the cylinder is completely submerged, the volume submerged in water and oil combined is equal to the volume of the cylinder.
Therefore, we can write: W_cylinder = W_fluid m_cylinder * g = (V_cylinder) * (density_water + density_oil) * g
Simplifying the equation: m_cylinder = (V_cylinder) * (density_water + density_oil)
Rearranging the equation to solve for the density of oil: density_oil = (m_cylinder / V_cylinder) - density_water
Plugging in the values: density_oil = (13.49 kg / 0.1605 m³) - 1000 kg/m³ ≈ 4.96 kg/m³
Now that we have the density of the oil, we can determine the height of the cylinder submerged in the oil.
Let's assume the height submerged in the oil is h_oil.
The volume of the cylinder submerged in the oil is given by V_oil = πr²h_oil.
The weight of the fluid displaced by the submerged portion is given by W_oil = V_oil * density_oil * g.
Since the weight of the cylinder is equal to the combined weight of the fluid displaced by the water and oil, we have: W_cylinder = W_water + W_oil
m_cylinder * g = (V_cylinder - V_oil) * density_water * g + V_oil * density_oil * g
Simplifying the equation: m_cylinder = (V_cylinder - V_oil) * density_water + V_oil * density_oil
Rearranging the equation to solve for the height of the oil: V_oil * (density_oil - density_water) = (m_cylinder - V_cylinder * density_water)
h_oil = (m_cylinder - V_cylinder * density_water) / (V_oil * (density_oil - density_water))
Plugging in the values: h_oil = (13.49 kg - 0.1605 m³ * 1000 kg/m³) / (π * (0.175 m)² * (4.96 kg/m³ - 1000 kg/m³))
Calculating the result, we find: h_oil ≈ 0.117 m
Therefore, approximately 0.117 m (or 11.7 cm) of the height of the cylinder is in the oil.