Question

Find the total pressure, in pascal, \(118 \mathrm{~m}\) below the surface of the ocean. The density of seawater is \(1.024 \mathrm{~g} / \mathrm{cm}^{3}\) and the atmospheric pressure at sea level is \(1.013 \times 10^{5} \mathrm{~Pa}\).

Short answer

The total pressure 118 m below the surface of the ocean is \(1287518.4 Pa\).

Step-by-step solution

Converting the units

We first need to convert the density of seawater from \(g / cm^3\) to \(kg / m^3\) as the other units in the formula are in the metric system. Since \(1g = 0.001kg\) and \(1cm^3 = 0.000001m^3\), the conversion gives \(\rho = 1.024 * 0.001 / 0.000001 = 1024 kg/m^3\).

Finding the pressure due to water

Next, we can find the pressure at the given depth due to the seawater using the formula \( P = \rho g h \). Let's use \(\rho = 1024 kg/m^3\), \( g = 9.8 m/s^2 \) and \( h = 118 m \). This gives the pressure as \( P = 1024 * 9.8 * 118 = 1184518.4 Pa \).

Finding the total pressure

The total pressure at the given depth in the ocean is the sum of the pressure due to the water and the atmospheric pressure at sea level. Given that \( P_0 = 1.013 * 10^{5} Pa \), the total pressure is \( P = 1184518.4 + 1.013 * 10^{5} = 1287518.4 Pa \).

Key concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air in the atmosphere on a surface area. At sea level, this pressure is about 101,300 Pascal (Pa). It plays a crucial role in fluid pressure problems as it is often a component of the total pressure experienced by a fluid at a certain depth.
This background pressure, present due to the air above, must be considered in scenarios involving underwater environments, such as oceans or lakes.

• Atmospheric pressure at sea level: approximately 101,300 Pa.
• Considered a constant in many fluid dynamics calculations at sea level.
• Added to the pressure due to fluid depth to find the total pressure.

Density Conversion

Density conversion is crucial for ensuring that all units in a calculation are consistent, particularly when dealing with liquids and pressures in different systems of measurement. In this example, the density of seawater is initially given in grams per cubic centimeter (g/cm³).
For pressure calculations, density should be in kilograms per cubic meter (kg/m³), which is the SI unit of density. This ensures compatibility with other SI units such as newtons and meters, often used in formulas to calculate pressure:

• 1 g/cm³ is equivalent to 1,000 kg/m³.
• Converting involves multiplying the g/cm³ value by 1,000 to get kg/m³.
• Ensures the hydrodynamic calculations are accurate and coherent.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. When calculating fluid pressure at a certain depth, hydrostatic pressure is key. It depends on three main factors: the density of the fluid (\(\rho\)), gravitational acceleration (\(g\)), and the height of the fluid column (\(h\)) above the point of measurement. The formula is:
\[P = \rho \cdot g \cdot h\]

• \(\rho\) represents fluid density in kg/m³.
• \(g\) is the acceleration due to gravity, approximately 9.8 m/s².
• \(h\) is the depth in meters.

This formula allows you to determine the pressure contribution from the water above, which is then combined with atmospheric pressure to find the total pressure."

Unit Conversion

Unit conversion is a fundamental process in physics that ensures the compatibility of different measurement units in calculations. Often, calculations involve combining inputs that use different unit systems, such as converting distances from centimeters to meters or masses from grams to kilograms.
For example, in fluid dynamics, we might need to convert:

• Grams to kilograms: Multiply by 0.001.
• Centimeters to meters: Multiply by 0.01.
• Ensures all variables in the formulas use consistent units.

This is essential when you are applying formulas that were derived using specific units, as failing to convert units correctly can lead to incorrect results and potentially hazardous miscalculations.