# Forces of Nature: Using Earth and Planetary Science for Teaching Physical Science

## Gravity: A Relatively Heavy Subject

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## Planetary Motion

Nearly everyone has heard about Newton and an apple. But few people seem to know the story behind it. Technically, there is no actual documentation for this story, so it might contain exaggerations. But it is relatively well accepted as having happened.

Prior to this incident, Newton had invented the Calculus, and with it had mathematically proven that an “inverse square law” dependence, such as gravitation on distance, must act as though all the mass of an object (the Earth) is at the exact center of the Earth.

Newton was trying to think of some way of experimentally confirming what he had already calculated, that inverse square dependence. He was sitting out in a field, looking at the Moon in the sky overhead. He believed that the Moon was orbiting the Earth because of the gravitation of the Earth. He believed that the Moon would normally have gone straight off into space, but the Earth’s gravitation caused it to “constantly fall” toward the Earth, making its path curved rather than straight. But he hadn’t thought of any way to experimentally prove that.

By his time, science had fairly accurately calculated the radius of the Earth, just under 4,000 miles (6,400 km). It was also known that the Moon orbited the Earth at an average distance of just under 240,000 miles (384,000 km, about 60 times as far from the center of the Earth as he was. These things were known.

When an apple fell from a tree near him, it suddenly dawned on him that the same Earth’s gravitation that must be curving the Moon’s path must also have made that apple accelerate toward the Earth in its fall.

His calculations had shown that the acceleration should not depend at all on the size or mass of the object. So, if that apple was out at the distance of the Moon, it should have the same acceleration as the Moon does, and would therefore also orbit the Earth. He knew that an apple falls at “the acceleration due to gravity”, 32 feet per second per second, what we call
**
g
**
. And that in the first second, that apple would fall very close to 16.1 feet (193”) toward the Earth.

Then, if that apple was moved to a place 60 times as far away from the center of the Earth, and gravitation actually did depend on an inverse square relationship, then the apple out there should fall 1/3600th as far as it did from the tree. So he multiplied 16.1 feet by 1/3600 and got an expected falling distance in one second to be 0.0535 inch.

That meant that the Moon must “fall” 0.0535” toward the Earth in a second (from an otherwise straight line. This is a small curvature (less than 1/16” over the 3,300 feet that the Moon moves every second!). But it turns out that it is still pretty easy to confirm. If you draw a really big circle that represents the orbit of the Moon, and then look at a small part of that circle, the part that the Moon moves through in one second, then simple geometry can determine that small curvature. (circle, chord, radius, etc.)

Interestingly, in this very simple calculation, the brilliant Newton apparently made a multiplication error regarding the radius of the Earth in inches! With this wrong value, there was no agreement in the results. Newton set aside this whole subject for six years! Around then, a new calculation of the radius of the Earth had been made (by Picard). Newton decided to try the calculation again, and he did it right this time, and the result was 0.0534”, a virtually perfect match. The inverse square law of gravitation was therefore proven. Also proven was the fact that the mass of the object, whether apple or Moon, did not affect the acceleration results.

As to this last statement, Newton later calculated that there actually IS a tiny effect due to the mass. But it is an extremely tiny effect, for any practical sized objects, because the Earth is so big and massive. There is also a tiny effect due to the differential gravitational effect of the Sun, which very slightly reduces the actual value for the Moon, which even explains that 0.0001” discrepancy.