# Density structure of the earth from mass and moment of inertia

Donald Forsyth
,
Brown University
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#### Summary

Using mass and moment of inertia as constraints, students construct density models of the whole earth. This activity uses concepts from physics, multi-dimensional calculus, and linear algebra, and also reinforces knowledge of density stratification.

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## Context

#### Audience

Introductory solid earth geophysics course, typically taken by junior or senior undergraduate concentrators or beginning graduate students. This activity is designed for a geophysics course.

#### Skills and concepts that students must have mastered

Multi-dimensional calculus
Physics - mechanics
Linear algebra

#### How the activity is situated in the course

Stand-alone exercise, although other exercises in seismology and gravity are related or use the results.

## Goals

#### Content/concepts goals for this activity

Density stratification of the earth; increasing density with depth and jump to much higher densities in core

#### Higher order thinking skills goals for this activity

Application of fundamental concept in physics to an earth science problem and using mathematical tools to solve the problem (3-D integration in spherical coordinates, solving simultaneous equations)

#### Other skills goals for this activity

Learn how to renormalize a set of linear equations to increase numerical accuracy.

Either working together as a group and/or overcoming fear of asking professor for advice is strongly encouraged.

## Description of the activity/assignment

After going over the concept of moment of inertia in class, students are asked to calculate density models of the earth. Three models of increasing complexity are developed, using additional constraints from seismology on the radius of the core. This homework assignment is typically the only assignment for the week - it gives the students practice in applying concepts and methods from physics and mathematics that usually have only been encountered in a theoretical fashion by the students. How we determine mass and moment of inertia are discussed in class. Usually students work together in small groups, as those whose math skills are rusty find this assignment difficult on their own. This activity uses geophysics to solve problems in other fields.

## Determining whether students have met the goals

Assignment is graded - if they haven't contributed to the group effort in solving the problem, it will show up on the midterm exam.

More information about assessment tools and techniques.