91Ó°ÊÓ

Pressure is a fundamental concept in fluid mechanics that helps us understand how forces work in fluids. In simple terms, pressure is defined as the force exerted over an area. It is expressed mathematically as:

• \[ P = \frac{F}{A} \]

where \( P \) is the pressure, \( F \) is the force, and \( A \) is the area over which the force is distributed.
In this exercise, we are dealing with two primary pressures: the atmospheric pressure and the hydrostatic pressure due to water at a certain depth.
Atmospheric pressure at sea level is approximately 101,325 pascals (Pa), which is equivalent to 1 atmosphere (atm). When calculating pressures underwater, we must consider both this atmospheric pressure and additional pressures due to the depth of water, which brings us to the concept of hydrostatic pressure.

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. In simpler words, it refers to the pressure felt at a specific depth in a fluid; this pressure increases with depth.The fundamental formula for calculating hydrostatic pressure is:

• \[ P_{\text{hydrostatic}} = \rho \cdot g \cdot h \]

where \( \rho \) (rho) represents the fluid's density, \( g \) is the acceleration due to gravity, and \( h \) is the depth below the surface.
In the given problem, we use the density of seawater, which is approximately 1025 kg/m³, the standard gravity rate of 9.81 m/s², and the depth of 30 meters to find the hydrostatic pressure.
This calculated pressure adds to the atmospheric pressure to give a total external pressure, exerted on the submersible's hatch. Clearly understanding these pressures allows us to move forward to see how they influence the force calculations needed to open the hatch.

Once we have understood the pressures involved, calculating the force exertion becomes straightforward. The goal is to find the amount of force needed to overcome both the pressure exerted on the hatch by the water above and the weight of the hatch itself.First, we calculate the net pressure on the hatch, which is the difference between the external pressure (sum of atmospheric and hydrostatic pressures) and the internal pressure inside the vehicle. This is expressed as:

• \[ \Delta P = P_{\text{external}} - P_{\text{internal}} \]

To find the force resulting from this net pressure, we multiply it by the area of the hatch (0.75 m²):

• \[ F_{\text{net}} = \Delta P \cdot A \]

Finally, since the crew also has to contend with the hatch's weight (300 N), the total downward force needed is the sum of this weight and the net force due to pressure difference:

• \[ F_{\text{crew}} = F_{\text{net}} + \text{Weight of the hatch} \]

This calculation shows that the total force required by the crew to open the hatch ensures safety during emergencies while dealing with complex underwater scenarios.