Page 2 of 4 in Universal Gravity
This text is meant to accompany class discussions. It is not everything there is to know about planetary motion. It is meant as a  prep for class. More detailed notes and examples are given in the class notes, presentations, and demonstrations (click here.)
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In addition to the 3 Laws of motion, Newton also came up with the the gravitational relationship between two objects. This relationship is know as "Newton's law of universal gravity." But before we get to Newton's work, let's look at what Newton knew about the Earth and Moon.


Earth's Radius and Cicumference

In 200 B.C. Eratosthenes, a greek mathematician and astronomer, estimated the radius of the Earth with some geometry. Eratosthenes noticed something unique and interesting in a well in Syene, which is now Aswan, Eqypt. He noticed that on June 21st when he looked down the well he saw an image of the Sun. He realized this meant the Sun was directly over head and the line connecting the sun and the well pointed to the center of the Earth. (See animation below.)

This never happened in his home town of Alexandria. Alexandria was about 473 miles to the North. (Actually Eratosthenes measured this distance to be about 5000 stades but the exact length of a stade in not known. It is estimated to be between 500 and 520 feet.) Eratosthenes also recognized that the Sun's ray where parallel when they hit the Earth.

On a 21st of June, when the sun is directly over Syene, Eratosthenes planted a slick vertically in the ground in Alexandria. He measured the angle between the top of the stick and the shadow.

The stick in Alexandria points to the center of the Earth. Eratosthenes saw the big picture and could figure out the circumference of the Earth by simular triangles.

By ratios, Eratosthenes figured out the circumference of the Earth.

He calculated the circumference of the Earth is about 23,000 miles. The actual radius is closer to 25,000 miles. Then he calculated the radius of the Earth. Using


Distance to the Moon

About 100 years later, in 100 BC, the greek Hipparchus estimated the distance between the Earth and the Moon. He determined where to stand so the Moon is directly overhead. This meant that a light ray from the Moon to this spot on the ground will go through the center of the Earth.

Next, Hipparchus, found a spot, where at this same time, the Moon is viewed on the horizon. He also measured the distance along the surface of the Earth between these two locations. This is called the arc length. This is the "arc" in the diagram above. The angle, in radians, is a ratio between the arc length and its angle and the circumference and its angle, (2πR).

Using trigonometry he calculated the distance from the surface of the Earth to the Moon. This calculation puts the Moon 59 times the radius of the Earth from the Earth's surface. This is 60 times the radius from the center of the Earth.

Aristarchus of Samos also found this distance about 150 years earlier. He used a different method. You can read about it here on Dr. Fowler's website. Look under the heading "How high is the Moon?"


by Tony Wayne ...(If you are a teacher, please feel free to use these resources in your teaching.)