Question

A submarine is designed to withstand an absolute pressure of 100 atm. How deep can it go below the water surface? (Consider the density of water = 1000 kg m⁻³, 1 atm = 1 $\times$ 10⁵ Pa and g = 10 m/s²)

• A. 990 m
• B. 9000 m
• C. 99 m
• D. 9900 m

Hint:

Always remember that "absolute pressure" includes the 1 atm from the air above. If you forget to subtract it, you would incorrectly calculate 1000 m.

Answer 1

The Correct Option is A

Step 1: Understanding the Topic:
This problem comes from "Mechanical Properties of Fluids." It focuses on hydrostatic pressure, which is the pressure exerted by a fluid at rest due to the force of gravity. A key concept here is "Absolute Pressure," which accounts for both the weight of the water above the object and the weight of the atmosphere above the water.
Step 2: Key Formulas and Approach:
The total or absolute pressure ($P_{abs}$) at a depth $h$ is: \[ P_{abs} = P_{atm} + \rho g h \] Where:
$P_{atm}$ is atmospheric pressure at the surface.
$\rho gh$ is the gauge pressure (pressure due only to the liquid).

Step 3: Detailed Explanation:

Identify given values: Max $P_{abs} = 100 \text{ atm}$. Since $1 \text{ atm} = 10^5 \text{ Pa}$, then $P_{abs} = 100 \times 10^5 \text{ Pa}$. Atmospheric pressure $P_{atm} = 1 \text{ atm} = 10^5 \text{ Pa}$.
Find Pressure from Water alone: The submarine already feels 1 atm of pressure at the surface. The water can only add $100 - 1 = 99 \text{ atm}$ before the limit is reached. \[ P_{water} = 99 \text{ atm} = 99 \times 10^5 \text{ Pa} \]
Calculate depth ($h$): Use the formula $P = \rho g h$: \[ 99 \times 10^5 = 1000 \times 10 \times h \] \[ 9,900,000 = 10,000 \times h \]
Solve: \[ h = \frac{9,900,000}{10,000} = 990 \text{ meters} \]
This means that for roughly every 10 meters you go down, you add 1 atm of pressure. 990 meters adds 99 atm to the 1 atm already at the surface, totaling 100 atm.
Step 4: Final Answer:
The submarine can go to a maximum depth of 990 m.