Question

Determine the pressure exerted on a diver at \(45 \mathrm{m}\) below the free surface of the sea. Assume a barometric pressure of \(101 \mathrm{kPa}\) and a specific gravity of 1.03 for seawater.

Short answer

Answer: The pressure exerted on the diver at 45 meters below the sea level is approximately 555.65 kPa.

Step-by-step solution

List and assess given information

The information we have is: - Depth of diver: \(45 \mathrm{m}\) - Barometric pressure: \(101 \mathrm{kPa}\) - Specific gravity of seawater: \(1.03\)

Convert barometric pressure to Pascals

Since the pressure we calculate will be in Pascals, we need to convert the given barometric pressure from kPa to Pascals: $$ 101\,\mathrm{kPa} \times 1000\,\mathrm{Pa/kPa} = 101000\,\mathrm{Pa} $$

Find the density of seawater

We can find the density of seawater using the specific gravity and the density of water (which is \(1000\, \mathrm{kg/m^3}\)): $$ \rho = \mathrm{Specific\,Gravity} \times \mathrm{Density\,of\,Water} = 1.03 \times 1000 = 1030\,\mathrm{kg/m^3} $$

Calculate the pressure due to depth

To find the pressure exerted due to the depth of the diver, we can use the formula: $$ P_\mathrm{depth}=\rho\times g\times h $$ where \(P_\mathrm{depth}\) = pressure due to depth \(\rho\) = density of seawater (\(1030\,\mathrm{kg/m^3}\)) \(g\) = acceleration due to gravity (\(9.81\,\mathrm{m/s^2}\)) \(h\) = depth of diver (\(45\,\mathrm{m}\)) Plugging in the values, we get: $$ P_\mathrm{depth} = 1030 \times 9.81 \times 45 = 454645.5\,\mathrm{Pa} $$

Calculate total pressure exerted on the diver

Now, we can find the total pressure exerted on the diver by adding the barometric pressure and the pressure due to depth: $$ P_\mathrm{total} = P_\mathrm{barometric} + P_\mathrm{depth} = 101000 + 454645.5 = 555645.5\,\mathrm{Pa} $$

Express the final answer in kPa

Now, let's convert the total pressure exerted on the diver back to kPa: $$ P_\mathrm{total} = 555645.5\,\mathrm{Pa} \times \frac{1\,\mathrm{kPa}}{1000\,\mathrm{Pa}} = 555.65\,\mathrm{kPa} $$ Hence, the pressure exerted on the diver at \(45 \mathrm{m}\) below the free surface of the sea is approximately \(555.65 \mathrm{kPa}\).

Key concepts

Specific Gravity

Understanding specific gravity is crucial when dealing with fluids, particularly when calculating the pressure underwater. Specific gravity is a dimensionless quantity that describes the ratio of the density of a substance to the density of a reference substance; typically, for liquids, the reference substance is fresh water. The formula to compute specific gravity (\text{SG}) is simply: \[ \text{SG} = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}} \]

Role of Specific Gravity in Pressure Calculations

To calculate the pressure underwater, knowing the specific gravity of the fluid where the object or person is submerged provides us with the fluid's density when multiplied by the density of water (nearly \(1000 \, \mathrm{kg/m^3}\) at standard conditions). This calculated density (\(\rho\)) is a key factor in determining the hydrostatic pressure exerted at any depth below the surface.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium at a given point within the fluid, due to the force of gravity. It increases in proportion to depth measured from the surface because of the increasing weight of the fluid exerting downward force from above. The equation for calculating hydrostatic pressure (\(P_{\mathrm{depth}}\)) at a depth (\(h\)) is:\[ P_{\mathrm{depth}}=\rho\times g\times h \]where\(\rho\) is the density of the fluid, and\(g\) is the acceleration due to gravity (\(9.81 \, \mathrm{m/s^2}\)).

Importance in Underwater Pressure

Calculating the hydrostatic pressure is essential for divers, marine engineers, and in a variety of other underwater activities, as it enables the determination of the force exerted by the water on objects at different depths.

Pascal's Principle

Pascal's principle, also known as Pascal's law, is a key concept in fluid mechanics, stating that when there is an increase in pressure at any point in a confined fluid, there is an equal increase in pressure at every other point in the container. It can be formalized as:\[ P_{\text{increase}} = \rho \times g \times h_{\text{increase}} \]where an increase in height (\(h_{\text{increase}}\)) relates to an increase in pressure (\(P_{\text{increase}}\)).

Application in Pressure Calculation

This principle explains why the pressure deep underwater is the same in all directions at a given depth and why it is only dependent on the depth and density of the fluid, not on the container's shape. As such, Pascal's principle is fundamental in determining the pressures exerted on submerged objects and is utilized in technologies such as hydraulic systems.

Barometric Pressure

Barometric pressure, also known as atmospheric pressure, is the force exerted upon a surface by the weight of the air above that surface in the Earth's atmosphere. It impacts various aspects of daily life, from weather forecasting to the calibration of pressure gauges.The standard atmospheric pressure at sea level is approximately \(101 \, \mathrm{kPa}\). However, barometric pressure varies with altitude and weather conditions and must be accounted for when calculating total pressure on a diver, as shown in our step-by-step solution.

Influence on Underwater Pressure Calculations

When adding the external atmospheric pressure to the hydrostatic pressure (pressure due to water depth), we get the total pressure exerted on an object. For underwater calculations, particularly at depths accessible by divers or submarines, the barometric pressure at the surface serves as the baseline to which the hydrostatic pressure is added to obtain the total pressure. This is key for understanding the complete pressure environment experienced by underwater entities.