Arizona State University COLLEGE OF LIBERAL ARTS AND SCIENCES
SCHOOL OF EARTH AND SPACE EXPLORATION

 

Mathematical Content: Basic algebra, formulae for gravitational attraction

Determining the Mass and Density of the Earth:

The goal of this study is to learn how the famous physicist, Sir Isaac Newton, computed the mass of the Earth, and then use this to compute its density.

Basic Concepts

 

In the year 1680, Sir Isaac Newton discovered the famous equation known as the Law of Gravitational Attraction on two objects. You will use this result, together with another result also due to Newton, to compute the mass of the Earth.

Notation:

Newton ’s Law of Gravitational Attraction, F = G ( m1m2 / r2), describes the force with which two objects attract each other.

  • G is called the universal gravitational constant and its value is known
  • m 1 and m 2 denote the masses of the two objects
  • r is the distance between the objects.

For an object with mass m on the surface of the earth, this equation becomes F = m (GME / RE2 ) where ME is the mass of the earth and RE is the radius of the earth.

Newton’s Second Law of Motion, F = mass x acceleration describes the relationship between force, mass and acceleration. On the surface of the earth the acceleration of gravity is about g = 9.8 m/s2 and we have F = mg .

Since each equation describes the same force, we have

m (GME / RE2 ) = Fgravity = mg, which simplifies to

(GME / RE2 ) = g

Thought questions

You will use the basic equation (GME / RE2 ) = g to estimate the mass, volume and density of the earth.

 

Information:

 

  • The average radius of the Earth is 6.38 x 106 meters
  • The universal gravitational constantG is G = 6.672 x 10-8m/(Mg·s2) where m is meters, Mg is megagrams (1 Mg = 1000 kilograms) and s is seconds.
  • T he acceleration of gravity at the surface of the Earth is about g = 9.8 m/s 2
  • The volume of a sphere of radius r is given by the formula V = 4/3 πr3
  • Density = mass / volume

 

The answers to the following questions will use Newton’s methods to determine the mass and density of the Earth.

Question 1: What is the mass of the Earth?

To answer this question, follow these steps:

  • Substitute the values given above in the equation (GME / RE2 ) = g.
  • Solve the equation for ME, the mass of the Earth.

Question 2 What is the volume of the Earth?

To answer this question, assume the Earth is a sphere with radius RE. Use the appropriate information given above.

 

Question 3 What is the density of the Earth?

Use your answers from the first two questions to answer this one.

 

Here are some follow-up questions to think about:

  1. How do you think Newton discovered his Law of Gravitational Attraction?
  2. Do you think this is an accurate method for computing the density of the Earth?
  3. Do you know of modern, more high-tech techniques that might give a more accurate result?
  4. How do you think geologists might measure the density of rocks?
  5. Why do you think it's important to know the density of the Earth, or rocks that are part of the Earth?
  6. What do you think the Earth is composed of to give it the density that you calculated?