A circular hatch in the hull of a submarine has a radius of 40.0 cm. The submarine is 850.0 m under water. If atmospheric pressure above the ocean is 1.01 x 105 Pa and the air pressure inside the submarine is 1.31x 105 Pa, what net force is exerted on the hatch? (ρsw = 1025 kg/m3)
Show your work.
To find the net force exerted on the hatch, we need to consider the pressure difference between the inside and outside of the submarine, as well as the area of the hatch.
The pressure difference is given by:
ΔP = Poutside - Pinside
Where Poutside is the atmospheric pressure above the ocean and Pinside is the air pressure inside the submarine.
We can calculate the pressure difference as:
ΔP = (1.01 x 10^5 Pa) - (1.31 x 10^5 Pa)
ΔP = -3.0 x 10^4 Pa
The negative sign indicates that the pressure inside the submarine is greater than the atmospheric pressure outside.
Next, we need to calculate the area of the hatch. Since it is circular, the area can be calculated using the formula:
A = πr^2
Where r is the radius of the hatch. Given that the radius is 40.0 cm, we convert it to meters:
r = 40.0 cm = 0.4 m
Now, we can calculate the area as:
A = π(0.4 m)^2 = 0.16π m^2
Finally, we can calculate the net force exerted on the hatch using the formula:
F = ΔP * A
Substituting the values, we get:
F = (-3.0 x 10^4 Pa) * (0.16π m^2)
Calculating this expression gives us the net force exerted on the hatch.
To determine the net force exerted on the hatch, we need to calculate the pressure difference between the interior and exterior of the hatch, and then multiply it by the area of the hatch.
Step 1: Calculate the pressure difference:
The pressure difference can be calculated using the equation:
△P = Pext - Pint
Where △P is the pressure difference, Pext is the pressure outside the submarine, and Pint is the pressure inside the submarine.
Given:
Pext = 1.01 x 105 Pa
Pint = 1.31 x 105 Pa
Substituting the values into the equation, we have:
△P = 1.01 x 105 - 1.31 x 105
△P = -0.30 x 105 Pa
Step 2: Convert the pressure difference into pascals:
Since 1 atm = 1.01 x 105 Pa, we can convert the pressure difference to pascals:
△P = -0.30 x 105 Pa * (1 atm / 1.01 x 105 Pa)
△P = -0.297 x 105 atm
Step 3: Convert the pressure difference to force:
To calculate the force, we'll use the equation:
F = △P * A
Where F is the force, △P is the pressure difference, and A is the cross-sectional area of the hatch.
Given:
Radius (r) = 40.0 cm = 0.4 m (converted to meters)
The area can be calculated using the equation:
A = π * r^2
Substituting the values into the equation, we have:
A = π * (0.4 m)^2
A = π * 0.16 m^2
A = 0.502 m^2
Now, we can calculate the net force:
F = -0.297 x 105 atm * 1.01 x 105 Pa/atm * 0.502 m^2
F = -0.150 x 105 N
The negative sign indicates that the net force is acting inwards, towards the interior of the submarine. Therefore, the magnitude of the net force exerted on the hatch is 0.150 x 105 N.