A submarine is stranded on the bottom of the ocean with its hatch
25.0 m below the surface. Calculate the force needed to open the hatch
from the inside, given it is circular and 0.450 m in diameter. Air pressure
inside the submarine is 1.00 atm.
To calculate the force needed to open the hatch, we need to consider the pressure difference between the inside and outside of the submarine.
1. Convert the depth of the hatch to meters:
The depth of the hatch is given as 25.0 m below the surface. Since pressure increases with depth in a fluid, we need to convert this depth to meters. There are 1,000 millimeters (mm) in a meter, so 25.0 m is equivalent to 25,000 mm.
2. Calculate the pressure difference:
The pressure difference (ΔP) between the inside and the outside of the submarine can be calculated using the formula:
ΔP = ρ * g * h
Where:
ρ is the density of the fluid (in this case, seawater) - approximately 1,000 kg/m³
g is the acceleration due to gravity - approximately 9.8 m/s²
h is the depth - in this case, 25,000 mm or 25.0 m
ΔP = 1000 kg/m³ * 9.8 m/s² * 25.0 m
3. Convert the result to pascals:
Since the pressure is typically measured in pascals (Pa), we need to convert the result to pascals. 1 pascal is equivalent to 1 Joule per square meter (J/m²).
Thus, the result from the previous step can be converted to pascals by multiplying it by 1,000. This is because the unit of pressure in atm is equivalent to approximately 101,325 Pa.
4. Calculate the force:
The force (F) needed to open the hatch can be calculated using the formula:
F = P * A
Where:
P is the pressure difference
A is the cross-sectional area of the hatch
The cross-sectional area (A) of a circular hatch can be calculated using the formula for the area of a circle:
A = π * (d/2)²
Where:
π is a constant approximately equal to 3.14159
d is the diameter of the hatch - in this case, 0.450 m
Now, we can substitute the values into the equation to calculate the force needed to open the hatch.