a huge oil tanker, salt water has flooded an oil tank to a depth of h2 = 5,0 m. On top of

Question

a huge oil tanker, salt water has flooded an oil tank to a depth of h2 = 5,0 m. On top of

the water is a layer of oil h1 deep, as in the cross-sectional view of the tank in Figure 1:

Figure 1
The oil has a density of 700 kg/m3
. If P0 is the atmospheric pressure and pressure at the
bottom, Pbot, is equal to 2,5 x 105 Pa, calculate the height of layer of oil, h1, at the top of
the water. (Take 1 025 kg/m3 as the density of salt water.) (5)
13. A force of 485 N is applied to the small piston of a hydraulic press whose area is 4 cm2
.
What is the minimum diameter of the large piston if it is to lift a 1960 N load? [𝐴 = 𝜋𝑟
2
]
(3)

Answer 1

To calculate the height of the layer of oil at the top of the water in the oil tanker, we can use the principle of hydrostatic pressure. The pressure at any point in a fluid is given by the formula P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid column.

In this case, we have two fluids - oil and salt water. The pressure at the bottom is given as Pbot = 2.5 x 10^5 Pa, which is the pressure at the bottom of the oil column. The pressure at the top is P0, which is the atmospheric pressure.

Let's calculate the pressure at the top of the water:
Pwater = P0 + ρwater * g * h2
Pwater = P0 + (1025 kg/m^3) * (9.8 m/s^2) * (5.0 m)

Now, let's calculate the pressure at the bottom of the oil column:
Poil = P0 + ρoil * g * (h1 + h2)
Poil = P0 + (700 kg/m^3) * (9.8 m/s^2) * (h1 + 5.0 m)

Since the pressure at the bottom of the oil column is given as Pbot = 2.5 x 10^5 Pa, we can set up an equation:
Pbot = P0 + (700 kg/m^3) * (9.8 m/s^2) * (h1 + 5.0 m)

Now, we can solve this equation to find the height of the oil layer, h1.

Moving the terms around, we get:
(700 kg/m^3) * (9.8 m/s^2) * (h1 + 5.0 m) = Pbot - P0

Dividing both sides by (700 kg/m^3) * (9.8 m/s^2), we get:
h1 + 5.0 m = (Pbot - P0) / (700 kg/m^3) * (9.8 m/s^2)

Subtracting 5.0 m from both sides, we get:
h1 = ((Pbot - P0) / (700 kg/m^3) * (9.8 m/s^2)) - 5.0 m

Now you can substitute the given values of Pbot = 2.5 x 10^5 Pa, P0 = atmospheric pressure, and solve for h1 to get the height of the oil layer at the top of the water.

For the second question, to find the minimum diameter of the large piston in a hydraulic press, we can use Pascal's law, which states that the pressure applied to a fluid in a confined space is transmitted undiminished to every portion of the fluid and to the walls of its container.

The hydraulic press consists of two pistons - a small piston with an area A1 and a large piston with an unknown area A2. The force applied to the small piston is given as 485 N, and the force required to lift the load is 1960 N.

According to Pascal's law, the pressure applied to the fluid is the same at both pistons:
P1 = P2

Pressure is defined as force divided by area:
P1 = F1 / A1
P2 = F2 / A2

Since P1 = P2, we can equate the two expressions:
F1 / A1 = F2 / A2

Now, we can rearrange this equation to solve for the unknown diameter of the large piston.

1. Convert the given area A1 from cm^2 to m^2:
A1 = 4 cm^2 * (1 m / 100 cm)^2

2. Substitute the given values of F1 = 485 N, F2 = 1960 N, and solve for A2:
F1 / A1 = F2 / A2

3. Rearrange the equation to solve for A2:
A2 = (F2 * A1) / F1

4. Since A = πr^2, we can find the radius of the large piston, and then double it to get the diameter:
r2 = √(A2 / π)
d2 = 2r2

Now you can substitute the given values of F1, F2, and A1, and solve for the minimum diameter of the large piston, d2.