A research submarine is operating at a depth of 115m in sea water of density 1025kg/m^3.
The absolute atmospheric pressure at sea level is 100kPa. The absolute air pressure inside the submarine is twice that at sea level. Find the net force exerted by the water outside and the air inside on a circular window of diameter 300m.
The idea is to calculate the pressure difference ΔP between the outside and the inside. Multiply ΔP by the area to get the force.
One pascal = 1 N-m-2.
Atmospheric pressure
Pa= 100 kPa
= 100*103 Pa
Density of sea water
= 1025 kg-m-3
= 1025*9.81 N-m-3
= 10.055*103 N-m-3
At 115m deep, hydrostatic pressure
Ph= 115*10.055*103 Pa
= 1156.3*103 Pa
Net pressure difference
=Atm.Press.+hydrost.pres.-internal press.
ΔP
= Pa+Ph-2Pa
=Ph-2Pa
=(1156.3-100)*103 Pa
=1056.3*103 Pa
Area of window (diameter 300 cm ?)
Aw= π*0.3²/4
=0.0707 m²
Force on window
= Aw*ΔP
= 0.0707 m² *1056.3*103 Pa
= 74669 N
To find the net force exerted by the water outside and the air inside on the circular window, we need to consider the pressure difference between the water and the air.
Step 1: Find the pressure at the depth of the submarine.
The pressure at a given depth in a fluid is given by the formula:
P = P₀ + ρgh
where:
P₀ is the pressure at the surface (sea level atmospheric pressure),
ρ is the density of the fluid (sea water density),
g is the acceleration due to gravity,
h is the depth.
Given:
- P₀ = 100 kPa = 100,000 Pa
- ρ = 1025 kg/m³
- g = 9.8 m/s²
- h = 115 m
Using the formula, we can find the pressure at the depth of the submarine:
P = 100,000 Pa + (1025 kg/m³ * 9.8 m/s² * 115 m)
P ≈ 100,000 Pa + 1,204,150 Pa
P ≈ 1,304,150 Pa
Step 2: Find the pressure inside the submarine.
The pressure inside the submarine is given as double the sea level atmospheric pressure:
P_submarine = 2 * 100,000 Pa
P_submarine = 200,000 Pa
Step 3: Calculate the net force on the window.
The difference in pressure between the water outside and the air inside the submarine is:
ΔP = P - P_submarine
ΔP = 1,304,150 Pa - 200,000 Pa
ΔP = 1,104,150 Pa
The net force on the window is given by:
F = ΔP * A
where:
ΔP is the pressure difference,
A is the area of the circular window.
Given:
- Diameter of the window, d = 300 m
We need to find the radius of the window (r = d/2) and then calculate the area (A = πr²).
Calculating the radius:
r = d/2 = 300 m/2 = 150 m
Calculating the area:
A = πr² = π * (150 m)² ≈ 70685.78 m²
Finally, we can calculate the net force exerted on the window:
F = ΔP * A
F ≈ 1,104,150 Pa * 70685.78 m² ≈ 77,945,518,903 N
Therefore, the net force exerted by the water outside and the air inside on the circular window is approximately 77,945,518,903 Newtons.