Problem 17 - Fundamentals of Physics

Question

Crew members attempt to escape from a dam-aged submarine \(100 \mathrm{~m}\) below the surface. What force must be applied to a pop-out hatch, which is \(1.2 \mathrm{~m}\) by \(0.60 \mathrm{~m},\) to push it out at that depth? Assume that the density of the ocean water is \(1024 \mathrm{~kg} / \mathrm{m}^{3}\) and the internal air pressure is at \(1.00 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The force required is approximately 650,996 Newtons.

Step 1: Understand the Problem

We need to find the force required to push out a submarine hatch at a depth of 100 meters. We're given the hatch dimensions, the density of ocean water, and internal air pressure. We need to consider both the water pressure outside and the atmospheric pressure inside the submarine.

Step 2: Calculate the Area of the Hatch

The hatch is rectangular, so calculate its area: \( \text{Area} = 1.2 \times 0.6 = 0.72 \ \text{m}^2 \).

Step 3: Determine the External Water Pressure

Water pressure at a depth can be calculated using \( P = \rho gh \), where \( \rho = 1024 \ \text{kg/m}^3 \), \( g = 9.81 \ \text{m/s}^2 \), and \( h = 100 \ \text{m} \). Thus, \( P = 1024 \times 9.81 \times 100 = 1,004,544 \ \text{Pa} \).

Step 4: Convert Internal Pressure to Pascals

Convert internal air pressure from atm to Pa. \( 1.00 \ \text{atm} = 101,325 \ \text{Pa} \).

Step 5: Calculate the Net Force Required

The net force is due to the difference between external water pressure and internal air pressure, multiplied by the hatch area. \[ F = (P_{\text{water}} - P_{\text{air}}) \times \text{Area} = (1,004,544 - 101,325) \times 0.72 \approx 650,996 \ \text{N} \].

Step 6: Finalize the Result

After calculating all necessary values, the force required to push out the hatch at 100 meters depth is approximately 650,996 Newtons.

Key Concepts

Force Calculation

A fundamental part of escaping from a submerged submarine involves understanding the force calculation needed to open the hatch. When a submarine is deep underwater, forces act on its surfaces from different directions. Specifically, the force that needs to be countered is due to the water pressure outside, compared to the internal atmospheric pressure inside the submarine.

The first step in force calculation involves determining the area of the hatch, which is rectangular. In this case, its area is calculated as the product of its length and width:

• Area = 1.2 meters (length) × 0.6 meters (width) = 0.72 square meters.

The force acting on an object is the product of pressure difference and the area over which that pressure acts. The external water pressure minus internal air pressure gives the pressure difference.

To find the net force, multiply this pressure difference by the area of the hatch:

• Net Force = (External Pressure - Internal Pressure) x Area
• Net Force = (1,004,544 Pa - 101,325 Pa) x 0.72 m²
• This results in a required force of approximately 650,996 Newtons.

This calculated force shows how significant the pressure of water can be compared to atmospheric pressure, especially at great depths.

Ocean Water Density

The density of ocean water is a crucial factor in determining the pressure exerted on objects submerged at great depths. Water density is generally given as the mass per unit volume, and ocean water density averages around 1024 kg/m³. This is slightly higher than that of fresh water due to the presence of salts and minerals in the ocean.

Density plays a vital role because the pressure exerted by a fluid in a stationary state increases linearly with depth. This relationship is defined by the formula:

• Pressure (\( P \)) = Density (\( \rho \)) × Gravitational acceleration (\( g \)) × Height (\( h \))

With gravity being constant at approximately 9.81 m/s², the pressure increases with each meter below the water surface. At a depth of 100 meters, the density of ocean water becomes a significant variable, leading to a high external pressure on the submarine hatch.

As density provides the basis for these calculations, understanding it helps to grasp the tremendous forces at play when a submarine is immersed in the ocean.

Submarine Hatch

A submarine hatch is designed for resilience against external pressures and is a critical point of focus when discussing force and hydrostatic pressure. These hatches must withstand tremendous pressure due to the surrounding water when submerged at significant depths. This makes the calculation of forces for opening them crucial.

Submarine hatches are built strong and often have mechanisms that rely on understanding precise pressure differences. The hatch dimensions directly affect the amount of force needed to operate them. Larger surfaces are typically subjected to greater forces because they offer more area for water pressure to act upon.

When the hatch's inside environment is maintained at 1 atm, while the outside is at a higher pressure due to water density and depth, it becomes vital to accurately calculate the necessary force to counter the water pressure. This accounts not only for the safety of the crew and the mechanical integrity of the hatch but also ensures that the submarine remains operational even under extreme conditions.

Hence being equipped with precise calculations and understanding the hatch's role in the overall architecture of the submarine is essential when planning maneuvers in aquatic environments.