Question

An average thickness of the oceanic crust is \(6 \mathrm{~km}\). Its density is \(2900 \mathrm{~kg} \mathrm{~m}^{-3}\). This is overlain by \(5 \mathrm{~km}\) of water \(\left(\rho_{w}=1000 \mathrm{~kg} \mathrm{~m}^{-3}\right)\) in a typical ocean basin. Determine the normal force per unit area on a horizontal plane at the base of the oceanic crust due to the weight of the crust and the overlying water.

Short answer

The pressure at the base of the oceanic crust is 219,312 kPa.

Step-by-step solution

Understand the Problem

The problem asks us to determine the normal force per unit area (or pressure) exerted at the base of the oceanic crust. This involves calculating both the pressure due to the weight of the oceanic crust and the overlying water.

Calculate the Pressure from Water

The pressure exerted by the water can be calculated using the formula for pressure due to a fluid column, which is given by \(P = \rho \cdot g \cdot h\), where \(\rho\) is the density of the fluid, \(g\) is the acceleration due to gravity \(\left(9.81 \mathrm{~m} / \mathrm{s}^2\right)\), and \(h\) is the height of the fluid column. For water, \(\rho = 1000 \mathrm{~kg/m^3}\) and \(h = 5 \mathrm{~km} = 5000 \mathrm{~m}\), so the pressure from water is:\[P_{water} = 1000 \times 9.81 \times 5000\]

Calculate the Pressure from Crust

Similarly, we calculate the pressure due to the crust. Using the same formula \(P = \rho \cdot g \cdot h\), for the crust \(\rho = 2900 \mathrm{~kg/m^3}\) and \(h = 6 \mathrm{~km} = 6000 \mathrm{~m}\), so the pressure from the crust is:\[P_{crust} = 2900 \times 9.81 \times 6000\]

Sum the Pressures

The total pressure at the base of the oceanic crust is the sum of the pressures from the water and the crust:\[P_{total} = P_{water} + P_{crust}\]

Calculate Total Pressure

Now, we compute each pressure. For water:\[\ P_{water} = 1000 \times 9.81 \times 5000 = 49050000 \mathrm{~Pa}\ (or \ 49050 \mathrm{~kPa})\]For crust:\[\ P_{crust} = 2900 \times 9.81 \times 6000 = 170262000 \mathrm{~Pa}\ (or \ 170262 \mathrm{~kPa})\]Thus, the total pressure is:\[\ P_{total} = 49050000 + 170262000 = 219312000 \mathrm{~Pa}\ (or \ 219312 \mathrm{~kPa})\]

Present the Final Solution

The normal force per unit area at the base of the oceanic crust due to the weight of the crust and overlying water is \( 219312 \mathrm{~kPa}\).

Key concepts

Oceanic Crust

The oceanic crust is a crucial part of our Earth's lithosphere, lying beneath the oceanic basins. It is thinner compared to continental crust. With an average thickness of about 6 kilometers, it consists mainly of basalt and gabbro. These rocks are resultants of volcanic processes.
Despite its thinness, the oceanic crust is relatively dense, with a density of approximately 2900 kg/m³. Its composition evolves through interactions with mantle material and water, playing a significant role in the plate tectonics cycle.

• Formation: Created at mid-ocean ridges where tectonic plates pull apart, allowing mantle material to rise and cool.
• Age: Oceanic crust is generally younger than continental crust, often only up to 200 million years old.
• Subduction: Oceanic crust can be recycled back into the mantle at subduction zones.

Understanding the properties of the oceanic crust can help geophysicists predict tectonic movements and the nature of earthquakes.

Pressure Calculations

Pressure calculations are essential in understanding how forces operate in geophysical scenarios such as the oceanic crust system. Pressure is the force exerted per unit area and is typically measured in Pascals (Pa) in the SI unit system. Calculating pressures involves knowing the density of materials involved, the height of the material column, and the acceleration due to gravity (9.81 m/s²).
In the context of the oceanic crust, calculations are relevant for determining the pressure exerted at the base of the crust by the overlying ocean water and the crust itself.

• Formula: \(P = \rho \cdot g \cdot h\)
• Factors: Density (\

Fluid Mechanics

Fluid mechanics helps us grasp how fluids behave and their various properties. It is an area of physics focusing on the forces and motions in fluids – both liquids and gases. In our context, it explains water's role on top of the oceanic crust.
For geophysicists, fluid mechanics offers insights into aspects like pressure distribution within the ocean. Since water is a fluid, pressure in a fluid column results from its weight. It increases with depth due to the gravitational force acting on the water's mass.

• Adds to Crust Pressure: Water's pressure on the crust can be calculated and added to the pressure exerted by the crust's own weight.
• Hydrostatic Pressure: Within a stationary fluid at rest, pressure increases linearly with depth.
• Applications: Understanding water pressure is vital in oceanic drilling operations and subsea infrastructure development.

This discipline not only aids in calculating forces but also contributes to predicting oceanic and atmospheric phenomena.

Crust Density

Crust density refers to the mass per unit volume of the oceanic crust. This characteristic helps in understanding crustal dynamics and its impact on Earth's surface movements.
In general, the oceanic crust's density is around 2900 kg/m³. This high density compared to the continental crust, which is less dense, causes the oceanic crust to often sit lower in the mantle, creating ocean basins. Density is affected by:

• Composition: The presence of basalt and other denser minerals increases overall density.
• Temperature and Pressure: Variations over depth and with surrounding settings can affect density measurement and behavior.
• Impact on Geodynamics: Higher density affects how oceanic plates are pushed or pulled under tectonic movements.

Understanding density plays a pivotal role in comprehending how and where stress within the Earth's crust develops.

Geodynamics

Geodynamics involves the study of the Earth's dynamics, particularly how processes within the Earth's interior cause movement and deformation of the crust. It is primarily concerned with the forces and motions that result from the interplay of density variations and pressure differences inside Earth.
Geodynamics applies these principles when examining how the oceanic crust interacts with the surrounding mantle and other tectonic plates. Factors important in geodynamics involve:

• Plate Tectonics: The constant movement of large plates of the Earth's crust above a convecting mantle.
• Subduction Zones: Areas where dense oceanic crust sinks back into the mantle.
• Earthquakes and Volcanoes: Direct results of geodynamic processes manifesting at the surface.

Geodynamics provides a framework that helps geophysicists understand and predict geological events, the Earth's structural formation, and the complex behaviors of crustal movements.