A submarine is operating at 100.0 m below the surface of the ocean . If the air inside the submarine is maintained at a pressure of 1.0 atmosphere , what is the magnitude of the force that acts on the rectangular hatch `2.0 m xx 1.0 `m on the deck of the submarine ?

Options

A. 980 N
B. `2. 0 xx 10^(6) N`
C. `2.0 xx 10^(3) N`
D. `9.8 xx 10^(5)` N

Answer

To solve the problem of finding the force acting on the rectangular hatch of the submarine, we can follow these steps:

### Step 1: Understand the pressures involved
The pressure inside the submarine is given as 1 atmosphere (P_inside = 1 atm). The pressure outside the submarine (P_out) can be calculated using the hydrostatic pressure formula, which accounts for the depth of the submarine in the ocean.

### Step 2: Calculate the pressure outside the submarine
The pressure outside the submarine can be calculated using the formula:
\[ P_{\text{out}} = P_{\text{atm}} + \rho g h \]
Where:
- \( P_{\text{atm}} \) = atmospheric pressure = 1 atm = \( 1.013 \times 10^5 \) Pa (approximately)
- \( \rho \) = density of seawater = \( 1000 \, \text{kg/m}^3 \)
- \( g \) = acceleration due to gravity = \( 9.81 \, \text{m/s}^2 \)
- \( h \) = depth = \( 100 \, \text{m} \)

Substituting the values:
\[ P_{\text{out}} = 1.013 \times 10^5 \, \text{Pa} + (1000 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(100 \, \text{m}) \]
\[ P_{\text{out}} = 1.013 \times 10^5 \, \text{Pa} + 981000 \, \text{Pa} \]
\[ P_{\text{out}} = 1.013 \times 10^5 \, \text{Pa} + 9.81 \times 10^5 \, \text{Pa} \]
\[ P_{\text{out}} = 1.0123 \times 10^6 \, \text{Pa} \]

### Step 3: Calculate the net pressure acting on the hatch
The net pressure acting on the hatch (P_net) is the difference between the outside pressure and the inside pressure:
\[ P_{\text{net}} = P_{\text{out}} - P_{\text{in}} \]
Where \( P_{\text{in}} = 1 \, \text{atm} = 1.013 \times 10^5 \, \text{Pa} \):
\[ P_{\text{net}} = 1.0123 \times 10^6 \, \text{Pa} - 1.013 \times 10^5 \, \text{Pa} \]
\[ P_{\text{net}} = 1.0123 \times 10^6 \, \text{Pa} - 0.1013 \times 10^6 \, \text{Pa} \]
\[ P_{\text{net}} = 0.911 \times 10^6 \, \text{Pa} \]

### Step 4: Calculate the area of the hatch
The area (A) of the hatch can be calculated as:
\[ A = \text{length} \times \text{width} = 2.0 \, \text{m} \times 1.0 \, \text{m} = 2.0 \, \text{m}^2 \]

### Step 5: Calculate the force acting on the hatch
The force (F) acting on the hatch can be calculated using the formula:
\[ F = P_{\text{net}} \times A \]
Substituting the values:
\[ F = 0.911 \times 10^6 \, \text{Pa} \times 2.0 \, \text{m}^2 \]
\[ F = 1.822 \times 10^6 \, \text{N} \]

### Final Answer
The magnitude of the force acting on the rectangular hatch is approximately:
\[ F \approx 1.82 \times 10^6 \, \text{N} \]

---