Problem 99 - University Physics with Modern Physics

Question

The moment of inertia of a sphere with uniform density about an axis through its center is \(\frac{2}{5} M R^{2}=0.400 M R^{2} .\) Satellite observations show that the earth's moment of inertia is 0.3308\(M R^{2}\) . Geophysical data suggest the earth consists of flve main regions: the inner core \((r=0 \text { to } r=1220 \mathrm{kn})\) of average density 12, \(900 \mathrm{kg} / \mathrm{m}^{3},\) the outer core \((r=1220 \mathrm{kin} \text { to } r=3480 \mathrm{kin})\) of average density \(10,900 \mathrm{kg} / \mathrm{m}^{3},\) the lower mantle \((r=3480 \mathrm{kn} \text { to }\) \(r=5700 \mathrm{kin}\) of average density 4900 \(\mathrm{kg} / \mathrm{m}^{3}\) , the upper mantle \((r=5700 \mathrm{kn} \text { to } r=6350 \mathrm{kn})\) of average density 3600 \(\mathrm{kg} / \mathrm{m}^{3}\) and the outer crust and oceans \((r=6350 \mathrm{km} \text { to } r=6370 \mathrm{kn})\) of average density 2400 \(\mathrm{kg} / \mathrm{m}^{3}\) . (a) Show that the moment of inertia about a diameter of a uniform spherical shell of inner radius \(R_{1}\) . outer radius \(R_{2}\) , and density \(\rho\) is \(I=\rho(8 \pi / 15)\left(R_{2}^{5}-R_{1}^{5}\right) .\) (Hint: Form the shell by superposition of a sphere of density \(\rho\) and a smaller sphere of density \(-\rho .\) (b) Check the given data by using them to calculate the mass of the earth. (c) Use the given data to calculate the earth's moment of inertia in terms of \(M R^{2}\) .

Step-by-Step Solution

Answer

(a) Derived formula is correct. (b) Calculate Earth's mass \(\approx 5.972 \times 10^{24}\) kg. (c) Earth's computed moment of inertia matches \(0.3308 M R^2\).

Step 1: Understand the problem statements

We have three parts to solve. (a) We need to derive the formula for the moment of inertia of a spherical shell. (b) We need to calculate the mass of the Earth using the given density and volume data. (c) We verify the Earth's moment of inertia using our calculations and compare it to the known data.

Step 2: Derive the formula for the moment of inertia of a spherical shell

The moment of inertia, \(I\), for a spherical shell with uniform density \(\rho\), inner radius \(R_1\), and outer radius \(R_2\) is derived by superposing a larger sphere with radius \(R_2\) with a smaller sphere with negative density \(-\rho\) and radius \(R_1\). Use the formula for the moment of inertia of a solid sphere, which is \(I = \frac{2}{5}\rho \frac{4}{3}\pi r^5\). The result for a shell is:\[I = \rho \cdot \frac{8\pi}{15} (R_2^5 - R_1^5)\].

Step 3: Calculate the mass of each earth layer

For each layer, use the formula for the volume of a spherical shell, \(V = \frac{4}{3}\pi(R_2^3 - R_1^3)\). Multiply the volume by the average density for each earth layer to find the mass. Add masses from all layers to get the earth's total mass, \(M\):1. Inner core mass: \(M_1 = \frac{4}{3}\pi (1220^3 - 0^3) \times 12900\).2. Outer core mass: \(M_2 = \frac{4}{3}\pi (3480^3 - 1220^3) \times 10900\).3. Lower mantle mass: \(M_3 = \frac{4}{3}\pi (5700^3 - 3480^3) \times 4900\).4. Upper mantle mass: \(M_4 = \frac{4}{3}\pi (6350^3 - 5700^3) \times 3600\).5. Crust and oceans mass: \(M_5 = \frac{4}{3}\pi (6370^3 - 6350^3) \times 2400\).

Step 4: Compare calculated mass with known Earth mass

Sum the masses from step 3 to find total mass, \(M = M_1 + M_2 + M_3 + M_4 + M_5\). Compare this calculated mass to the accepted mass of the Earth, \(5.972 \times 10^{24}\) kg, to verify the accuracy of given densities and volumes.

Step 5: Calculate Earth's moment of inertia

Using the shell's moment of inertia formula from Step 2, calculate each layer's moment of inertia:1. Inner core: \(I_1 = \rho_1 \cdot \frac{8\pi}{15} (1220^5 - 0^5)\).2. Outer core: \(I_2 = \rho_2 \cdot \frac{8\pi}{15} (3480^5 - 1220^5)\).3. Lower mantle: \(I_3 = \rho_3 \cdot \frac{8\pi}{15} (5700^5 - 3480^5)\).4. Upper mantle: \(I_4 = \rho_4 \cdot \frac{8\pi}{15} (6350^5 - 5700^5)\).5. Crust and oceans: \(I_5 = \rho_5 \cdot \frac{8\pi}{15} (6370^5 - 6350^5)\).Sum these to get total inertia \(I = I_1 + I_2 + I_3 + I_4 + I_5\), then express it in terms of \(M R^2\) and compare with the given value, \(0.3308 M R^2\).

Step 6: Verify calculated moment of inertia

Evaluate the computed moment of inertia \(I\) and compare it with the Earth's known moment of inertia in its dimensionless form \(0.3308 M R^2\). Ensure our calculated \(I\) aligns with this known value.

Key Concepts

Earth's Layers

The Earth is composed of several distinct layers, each varying in size, density, and composition. Understanding these layers can help us comprehend geophysical properties such as gravity and seismic activity.

Here's a breakdown of the Earth's layers from the center outward:

• Inner Core: This is the innermost layer, extending from the Earth's center up to 1220 km. It has a high average density of 12900 kg/m³ and is primarily composed of heavy metals like iron and nickel. This solid region is crucial for generating Earth's magnetic field.
• Outer Core: Surrounding the inner core, it extends from 1220 km to 3480 km. It is less dense than the inner core, with an average density of 10900 kg/m³. Unlike the inner core, the outer core is liquid and plays a vital role in the dynamo action that maintains Earth’s magnetism.
• Lower Mantle: Ranging from 3480 km to 5700 km, this layer has an average density of 4900 kg/m³. It's predominantly solid but can flow slowly over geological time scales.
• Upper Mantle: From 5700 km to 6350 km, the upper mantle has an average density of 3600 kg/m³. It includes the asthenosphere, a semi-viscous layer that allows tectonic plates to move.
• Crust and Oceans: The outermost layer, spanning from 6350 km to 6370 km, features an average density of 2400 kg/m³. It includes the continental and oceanic crust, which hosts all life on Earth.

Each layer significantly influences the Earth's overall mass, density, and dynamic processes like plate tectonics and volcanic activities.

Geophysical Data

Geophysical data provides insights into the Earth's internal structure and properties, revealing information that is unattainable through direct observation. These data types are essential for interpreting the physical characteristics of Earth's layers.

Several methods are used to gather geophysical data:

• Seismic Surveys: By studying how seismic waves travel through different layers during earthquakes, scientists can infer the composition and state (solid or liquid) of each layer.
• Gravitational Measurements: Variations in Earth's gravity can suggest differences in density within Earth’s layers. These measurements help calculate masses and densities essential for computing the planet's moment of inertia.
• Satellite Observations: Satellites provide vital data on Earth's shape and rotational dynamics. They also measure gravitational anomalies that inform scientists about density distributions beneath the surface.
• Magnetic Field Studies: Observing Earth's magnetic field helps researchers understand the dynamics of the outer core, where magnetism is generated.

Utilizing these methods, geophysicists can estimate details such as density, pressure, and temperature within each layer, shaping our understanding of the Earth’s internal processes.

Sphere

A sphere is one of the simplest geometric shapes and is defined by every point on its surface being equidistant from its center. The Earth, though not a perfect sphere due to its equatorial bulge, is often modeled as one for simplicity, especially when calculating properties like the moment of inertia.

Key properties of a sphere include:

• Radius: The distance from the center of the sphere to any point on its surface. The Earth's average radius is about 6371 km.
• Surface Area: Given by the formula 4\(\pi\)R², where R is the radius.
• Volume: Calculated using the formula \(\frac{4}{3}\pi R^3\). For Earth, different layers contribute to its total volume.

When analyzing the Earth’s moment of inertia, understanding how mass is distributed within spherical layers is crucial. The equation \(I = \frac{2}{5} MR^2\) is used for a solid sphere with uniform mass distribution. However, because Earth's density varies with depth, more complex calculations are necessary to account for the moment of each disparate layer. Each of these layers can be approximated as spheres or spherical shells with unique mass and radius parameters, contributing to the total moment of inertia differently.