Question

The lower half of a 6 -m-high cylindrical container is filled with water \(\left(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\right)\) and the upper half with oil that has a specific gravity of \(0.85 .\) Determine the pressure difference between the top and bottom of the cylinder.

Short answer

Question: Determine the pressure difference between the top and the bottom of a 6 m tall cylinder, with the top half filled with oil (specific gravity 0.85) and the bottom half filled with water. Answer: The pressure difference between the top and the bottom of the cylinder is 54,435.5 Pa or N/m^2.

Step-by-step solution

Identify the Parameters

The container's total height is 6 m, with half (i.e., 3 m) filled with water and half filled with oil. The water's density is \rho_w = 1000 kg/m^3 and the oil's specific gravity is 0.85. The specific gravity is the ratio of the density of a substance to the density of a reference substance (water for liquids), so we get oil's density as ρ_o = 0.85ρ_w = 0.85 * 1000 = 850 kg/m^3. Gravitational acceleration is denoted by g, where g = 9.81 m/s^2.

Calculate Pressure Due To Water

Using the formula for pressure in fluids, P = ρgh, the pressure due to water, P_w, is calculated as P_w =ρ_w * g * h. Substituting the known values: P_w = 1000 kg/m^3 * 9.81 m/s^2 * 3 m = 29430 Pa or N/m^2.

Calculate Pressure Due To Oil

Similarly, the pressure due to oil, P_o, can be calculated by using the formula P = ρgh but taking the oil's density. Substituting the known values gives: P_o = 850 kg/m^3 * 9.81 m/s^2 * 3 m = 25005.5 Pa or N/m^2.

Calculate the Pressure Difference

The pressure difference between the top and the bottom of the cylinder is given by the sum of the pressures due to the water and the oil. Hence, pressure difference = P_w + P_o = 29430 Pa + 25005.5 Pa = 54435.5 Pa or N/m^2.