A “U” shaped tube (with a constant radius) is filled with water and oil as shown. The water is a height h1 = 0.35 m above the bottom of the tube on the left side of the tube and a height h2 = 0.12 m above the bottom of the tube on the right side of the tube. The oil is a height h3 = 0.3 m above the water. Around the tube the atmospheric pressure is Patm = 100 kPa. Water has a density of 103 kg/m3.
What is the density of the oil?
To find the density of the oil, we can use the principle of Pascal's law which states that the pressure is the same at all points on the same horizontal plane within a continuous fluid.
First, we need to determine the pressure at the interface between the water and the oil on the left side of the tube. This can be calculated using the hydrostatic pressure formula:
P_water = ρ_water * g * h1
P_oil = P_water + ρ_oil * g * h3
Since the pressure is the same at the interface between the water and the oil on the left and right side of the tube, we have:
P_water = P_oil
ρ_water * g * h1 = (ρ_water * g * h3) + (ρ_oil * g * h3)
Substitute the known values:
103 kg/m^3 * 9.81 m/s^2 * 0.35 m = (103 kg/m^3 * 9.81 m/s^2 * 0.3 m) + (ρ_oil * 9.81 m/s^2 * 0.3 m)
Solve for the density of the oil:
36.0435 = 29.43 + 2.943 * ρ_oil
6.6135 = 2.943 * ρ_oil
ρ_oil ≈ 2.25 kg/m^3
Therefore, the density of the oil is approximately 2.25 kg/m^3.