Problem 23 - Fundamentals of Physics

Question

In analyzing certain geological features, it is often appropriate to assume that the pressure at some horizontal level of compensation, deep inside Earth, is the same over a large region and is equal to the pressure due to the gravitational force on the overlying material. Thus, the pressure on the level of compensation is given by the fluid pressure formula. This model requires, for one thing, that mountains have roots of continental rock extending into the denser mantle Consider a mountain of height \(H=6.0 \mathrm{~km}\) on a continent of thickness \(T=32 \mathrm{~km} .\) The continental rock has a density of \(2.9 \mathrm{~g} / \mathrm{cm}^{3}\), and beneath this rock the mantle has a density of \(3.3 \mathrm{~g} / \mathrm{cm}^{3} .\) Calculate the depth \(D\) of the root. (Hint: Set the pressure at points \(a\) and \(b\) equal; the depth \(y\) of the level of compensation will cancel out.)

Step-by-Step Solution

Answer

The depth of the root is 43.5 km.

Step 1: Understand the given parameters

We have a mountain of height \( H = 6.0 \text{ km} \), a continental thickness \( T = 32 \text{ km} \), density of continental rock \( \rho_c = 2.9 \text{ g/cm}^3 \), and the density of the mantle \( \rho_m = 3.3 \text{ g/cm}^3 \). Our goal is to find the depth \( D \) of the mountain's root.

Step 2: Express pressure at compensation level

The pressure at the level of compensation due to the mountain with the root is equal to the pressure due to the mantle without the mountain. Let's find expressions for these two pressures: - Pressure due to the mountain with the root: \( P_1 = \rho_c \cdot g \cdot (H + D) \).- Pressure due to the mantle alone: \( P_2 = \rho_m \cdot g \cdot D \). Set \( P_1 = P_2 \) because pressures at the level of compensation are equal.

Step 3: Set equations equal and solve for D

Equating pressures, we have: \[ \rho_c \cdot g \cdot (H + D) = \rho_m \cdot g \cdot D \] The gravitational constant \( g \) cancels out from both sides, leading to: \[ \rho_c \cdot (H + D) = \rho_m \cdot D \]

Step 4: Rearrange and solve the equation

We rearrange the equation from Step 3 to solve for \( D \): \[ \rho_c \cdot H + \rho_c \cdot D = \rho_m \cdot D \] \[ \rho_c \cdot H = \rho_m \cdot D - \rho_c \cdot D \] \[ \rho_c \cdot H = (\rho_m - \rho_c) \cdot D \] Solving for \( D \), we find: \[ D = \frac{\rho_c \cdot H}{\rho_m - \rho_c} \]

Step 5: Calculate the depth of the root

Substitute the known values: \( \rho_c = 2.9 \text{ g/cm}^3 \), \( \rho_m = 3.3 \text{ g/cm}^3 \), and \( H = 6.0 \text{ km} = 6000 \text{ m} \): \[ D = \frac{2.9 \cdot 6000}{3.3 - 2.9} \] Calculate: \[ D = \frac{17400}{0.4} = 43500 \text{ m} = 43.5 \text{ km} \] Thus, the depth of the root is \( 43.5 \text{ km} \).

Key Concepts

Understanding Isostasy

Isostasy is like Earth's own balancing act. It's a concept in geophysics explaining how different parts of the Earth's crust float at various heights due to their densities and thicknesses. Imagine a seesaw with weights on both ends seeking balance. The Earth, in a similar manner, maintains balance by having less dense continental crust rise above the denser oceanic crust.
The concept of isostasy helps us understand why mountains have roots. These roots extend into the deeper, denser mantle, akin to how a floating iceberg has most of its volume beneath the water surface. Isostatic balance is essential for forming and maintaining Earth's varying topographical features, like mountains and valleys.

• Explains continental elevation variations
• Describes floating crust in gravitational equilibrium
• Analogy: Mountains have deep roots like icebergs

This understanding of isostasy allows geophysicists to predict how Earth's crust will respond to changes, such as ice caps melting or tectonic shifts. It's like knowing where the seesaw will tilt when a new weight is added.

Principles of Pressure Calculation

Pressure calculation, especially in geological contexts, involves understanding how forces exert on materials at certain depths. In the Earth’s crust, pressure is primarily due to the weight of the overlying rock. The deeper we go, the more rock there is above, hence more pressure.
In the context of isostasy, pressure calculation becomes crucial. We equate pressures at a compensation depth to visualize how mountains float atop a denser mantle layer. This is seen in the exercise through the pressure equilibrium between mountain roots and the surrounding mantle.
The pressure at any depth under the Earth is defined by the formula:\[ P = ho imes g imes h \]Where:

• \( P \) is pressure
• \( \rho \) is density
• \( g \) is the gravitational acceleration
• \( h \) is depth

Knowing the density and thickness of materials helps us calculate the pressure at any point. This understanding allows not only for the examination of mountain formations but also for the prediction of geological activities.

Interconnection of Density and Gravity

Density and gravity play vital roles in understanding isostasy and pressure calculations in geophysics. Density is the amount of mass per unit volume. Different types of rocks and materials in Earth's crust have distinctly different densities. For instance, continental rocks are less dense than mantle material, explaining why continents float at a higher elevation.
The gravitational force pulls denser materials more strongly. Thus, gravity is a critical factor in controlling how various layers of Earth behave. In our exercise, you saw how the difference in density between the continental and mantle rock led to a significant root depth. Mathematically, the difference in density defines how towering a mountain can be when considering its root.
In formulas and calculations, gravity is often symbolized by \( g \), typically approximating 9.8 m/s² on Earth's surface. This gravitational constant helps us calculate pressure and understand the distribution of materials within the Earth based on their density. These calculations using density and gravity help create accurate models of the Earth's subsurface.

• Density: Mass per volume
• Gravity: Force affecting layers
• Interplay governs topographical features

Understanding these concepts is key to grasping the principles of isostasy and pressure, enabling us to analyze Earth's structure more comprehensively.