oil = 0.88 m
brine= 1.11 m
P oil= 790 kg/m^3
P brine= 1025 kg/m^3
What is the gauge pressure at the bottom of the cylinder?
Using the gauge pressure, calculate the force exerted at the bottom of cylinder tube whose radius is 90 cm.
What is the absolute pressure at the bottom of the cylinder considering that the container is open on top?
To calculate the gauge pressure at the bottom of the cylinder, we need to find the difference between the absolute pressure at the bottom and atmospheric pressure.
The absolute pressure at the bottom of the cylinder can be calculated using the hydrostatic pressure 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.
For the oil, we have:
P_oil = ρ_oil * g * h_oil
= 790 kg/m^3 * 9.8 m/s^2 * 0.88 m
= 6829.44 N/m^2
For the brine, we have:
P_brine = ρ_brine * g * h_brine
= 1025 kg/m^3 * 9.8 m/s^2 * 1.11 m
= 11007.15 N/m^2
To calculate the gauge pressure, we subtract atmospheric pressure (which is typically around 101,325 N/m^2):
Gauge pressure_oil = P_oil - atmospheric pressure
= 6829.44 N/m^2 - 101,325 N/m^2
= -94695.56 N/m^2
Gauge pressure_brine = P_brine - atmospheric pressure
= 11007.15 N/m^2 - 101,325 N/m^2
= -90317.85 N/m^2
Note that the negative sign indicates that the pressure is below atmospheric.
To calculate the force exerted at the bottom of the cylinder, we use the formula:
Force = Pressure * Area
The radius of the cylinder tube is given as 90 cm, which is equal to 0.9 m. The area can be calculated using the formula:
Area = π * r^2
Area = π * (0.9 m)^2
= 2.54 m^2
Force_oil = Gauge pressure_oil * Area
= -94695.56 N/m^2 * 2.54 m^2
= -240,654.24 N
Force_brine = Gauge pressure_brine * Area
= -90317.85 N/m^2 * 2.54 m^2
= -229188.99 N
Again, the negative sign indicates that the force is directed upward.
To find the absolute pressure at the bottom of the cylinder when the container is open on top, we need to consider atmospheric pressure. The absolute pressure is the sum of the gauge pressure and atmospheric pressure:
Absolute pressure_oil = Gauge pressure_oil + atmospheric pressure
= -94695.56 N/m^2 + 101,325 N/m^2
= 6630.44 N/m^2
Absolute pressure_brine = Gauge pressure_brine + atmospheric pressure
= -90317.85 N/m^2 + 101,325 N/m^2
= 10907.15 N/m^2