Question

Consider a U-tube whose arms are open to the atmosphere. Now water is poured into the U-tube from one arm, and light oil \(\left(\rho=790 \mathrm{kg} / \mathrm{m}^{3}\right)\) from the other. One arm contains 70 -cm-high water, while the other arm contains both fluids with an oil-to-water height ratio of \(4 .\) Determine the height of each fluid in that arm.

Short answer

Answer: In the arm of the U-tube containing both fluids, the height of the oil is 0.886 m and the height of the water is 0.2215 m.

Step-by-step solution

Apply the hydrostatic pressure principle

At the interface between the water and oil layers, the hydrostatic pressures from both arms of the U-tube must be equal since both arms are exposed to atmospheric pressure. \(p_{water} = p_{oil}\) Here, the pressure exerted by water and oil will be given by their heights and densities, considering g as the gravitational acceleration. \(\rho_{water} \cdot h_{water} \cdot g = \rho_{oil} \cdot h_{oil} \cdot g\)

Use the given data to simplify the equation

We know the density of oil (\(\rho_{oil}\)), the height of water in one arm (70 cm), and the oil-to-water height ratio (\(\frac{h_{oil}}{h_{water}} = 4\)). Also, let's convert the height of water to meters (0.7 m), and use the density of water as 1000 kg/m³. Substitute these values into the equation: \(1000 \cdot 0.7 \cdot g = 790 \cdot h_{oil} \cdot g\) Since both sides of the equation contain g, it cancels out: \(1000 \cdot 0.7 = 790 \cdot h_{oil}\)

Solve for the height of oil

Now, we can solve the equation to find the height of oil in the U-tube: \(h_{oil} = \frac{1000 \cdot 0.7}{790} = 0.886 \mathrm{m}\)

Use the oil-to-water height ratio to find the height of water

We are given the oil-to-water height ratio as 4. Now we can find the height of water in that arm: \(h_{water} = \frac{h_{oil}}{4} = \frac{0.886}{4} = 0.2215 \mathrm{m}\) Now, we have determined the height of each fluid in the arm with both fluids: the height of the oil is 0.886 m and the height of the water is 0.2215 m.