Question

Determine the pressure exerted on the surface of a submarine cruising \(175 \mathrm{ft}\) below the free surface of the sea. Assume that the barometric pressure is 14.7 psia and the specific gravity of seawater is 1.03

Short answer

Answer: The pressure exerted on the surface of the submarine cruising at 175 ft below the free surface of the sea is approximately 640.32 kPa.

Step-by-step solution

Calculate hydrostatic pressure

To find the hydrostatic pressure, we need to first convert the depth from feet to meters. We'll use the conversion factor 1 meter = 3.281 feet. Depth in meters = \(\frac{175 \, ft}{3.281 \, ft/m} \approx 53.34 \, m\) Now, we'll use the specific gravity of seawater and the specific weight of water to find the specific weight of seawater. The specific weight of water is approximately 9.81 kN/m³. Specific weight of seawater = Specific gravity × Specific weight of water = 1.03 × 9.81 kN/m³ \(\approx\) 10.10 kN/m³ Now, we can calculate the hydrostatic pressure using the formula: Hydrostatic pressure = Specific weight of seawater × Depth = 10.10 kN/m³ × 53.34 m \(\approx\) 539 kPa

Convert barometric pressure

The given barometric pressure is in psia (pounds per square inch absolute), but we need to convert it to kPa to match the unit of hydrostatic pressure. 1 psi \(\approx\) 6.895 kPa Barometric pressure in kPa = 14.7 psia × 6.895 kPa/psi \(\approx\) 101.32 kPa

Calculate total pressure exerted on the submarine

The total pressure is the sum of hydrostatic pressure and barometric pressure. Total Pressure = Hydrostatic pressure + Barometric pressure Total Pressure = 539 kPa + 101.32 kPa \(\approx\) 640.32 kPa The pressure exerted on the surface of the submarine cruising at 175 ft below the free surface of the sea is approximately 640.32 kPa.

Key concepts

Specific Gravity

Specific gravity is a dimensionless unit that describes the density of a substance compared to the density of a reference substance, typically water for liquids and solids. It's calculated by dividing the density of the substance by the density of the reference substance at a specified temperature, usually 4°C for water because that's when it's at its densest. For our submarine scenario, the specific gravity of seawater is given as 1.03. This means seawater is 1.03 times as dense as freshwater.

Understanding specific gravity is crucial when dealing with buoyancy, hydrostatic pressure, and fluid mechanics because it can impact how objects behave when submerged in different liquids.

Specific Weight

Specific weight, also known as unit weight, is the weight per unit volume of a material. The standard unit is typically Newtons per cubic meter (N/m³) or kiloNewtons per cubic meter (kN/m³). It is calculated by multiplying the density of the material by the acceleration due to gravity. In the example of the submarine, the specific weight of seawater is found by multiplying the specific gravity of seawater by the specific weight of water (approximately 9.81 kN/m³), resulting in approximately 10.10 kN/m³.

For objects submerged in fluids, specific weight helps determine the hydrostatic pressure exerted by the fluid at depth, making it a critical factor in many engineering and physics calculations.

Pressure Conversion

When working with pressure, one may encounter different units such as Pascals (Pa), kiloPascals (kPa), pounds per square inch (psi), and atmospheres (atm). To compare pressures or add them together as in our exercise, one must convert all pressures to the same units. The conversion from psi to kPa is done using the factor 1 psi is approximately equal to 6.895 kPa. Conversions like these are essential when solving problems that include different pressure measurements.

This step is fundamental because pressure calculations are common in various scientific and engineering disciplines, including fluid mechanics, meteorology, and medicine. In our case, converting barometric pressure from psia (pounds per square inch absolute) to kPa allowed for an accurate summation with hydrostatic pressure to determine the total pressure exerted on the submarine.

Barometric Pressure

Barometric pressure, also known as atmospheric pressure, is the force per unit area exerted by the weight of the atmosphere at a given point. It is measured by a barometer and is often reported in inches of mercury (Hg), millibars (mb), or Pascals (Pa). One standard atmosphere (atm) is equivalent to 101.325 kPa, or about 14.7 psi, which represents the average atmospheric pressure at sea level.

The value of barometric pressure is vital when working with gases, weather forecasting, and when calculating the total pressure experienced by objects, such as submarines, under the sea surface. Barometric pressure varies with altitude and weather conditions but knowing its value at sea level, as provided in our exercise, is a key component to total pressure calculations for submerged objects.