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Saturn from Cassini Astronomy 161:
An Introduction to Solar System Astronomy
Prof. Richard Pogge, MTWThF 2:30

Lecture 18: The Apple and the Moon:
Newtonian Gravity

Key Ideas:

Law of Falling Bodies (Galileo) Law of Universal Gravitation (Newton)
NOTE:
This and the following lecture are probably the most mathematical of all the lectures that will be given in this class. I encourage you all to read these notes in advance and try to follow the arguments in them. In will make it easier to follow along during lecture. [rwp]

The Law of Falling Bodies

Prior to his telescopic work, Galileo performed fundamental research on motion.

Explored the rate of falling bodies by dropping different weights, or sliding them down inclined planes.

Law of Falling Bodies


Universal Mutual Gravitation

Isaac Newton, in his Principia, formulated the Law of Universal Mutual Gravitation:
Gravity is an Attractive force:
It draws massive objects closer together

Gravity is a Universal force:
It operates everywhere in the Universe.

Gravity is a Mutual force:
It works between pairs of massive objects.

Gravitational Force

The Force of Gravity between any two objects depends only upon:

The masses of the two objects:

The distance between them:

It does not depend on the shapes, colors, or compositions of the objects.


The Law of Universal Gravitation

The force of gravitational attraction between any two massive bodies is proportional to their masses and inversely proportional to the square of the distance between their centers.
The Force of Gravity is an example of an Inverse Square Law Force

Stated mathematically, the force of gravity between two massive bodies is:

F=GM1M2/d^2

Where:
F = force due to gravity.
M1 = mass of the first body
M2 = mass of the second body
d = distance between their centers.
G = Gravitational Force Constant

The Gravitational Force Constant

The force constant, G, is a number which gives the size of the gravitational coupling between two massive objects.

G is very small, in metric units:

G=6.7x10-11 Newtons meter2 / kilogram2
The Newton is the metric unit of force:
4.41 Newtons = 1 pound
G has to be measured experimentally18.1.

The Fall of an Apple.

Stand on the Earth and drop an apple.

What is the force of the Earth on the apple?

F = GMearth Mapple/Rearth2

What is the apple's acceleration (Newton's 2nd Law of Motion):

aapple = F/Mapple = GMearth/Rearth2 = 9.8 meters/sec2
Note that the mass of the apple (Mapple) had divided out of the equation. This means that the acceleration due to gravity is independent of the mass of the apple, just like Galileo had shown earlier.

Equal and Opposite Reactions

But, Newton's Third Law of Motion states that all forces come in equal yet opposite pairs

What force does the the apple apply in return upon the Earth?

F = GMearth Mapple/Rearth2

How much does the Earth accelerate towards the apple?

aearth = F/Mearth = GMapple/Rearth2
This can be rewritten to give the acceleration of the Earth in terms of the acceleration of the apple towards the Earth as
aearth = aapple x (Mapple/Mearth)
where aapple=9.8 meters/sec2, and the ratio of the Mass of the apple to the Mass of the Earth is very small number. For a typical 200g apple, this works out to be about 10-25 meters/sec2, a very tiny acceleration.

The Mass of the Earth

We can directly measure the acceleration of gravity at the surface of the Earth by dropping objects and timing their fall (e.g., like was done by Galileo). We find
a = 9.8 meters/sec2

We can also measure the radius of the Earth using geometry (Eratosthenes):

Rearth=6378 kilometers = 6,378,000 meters

Combining these together using Newton's formula for the Gravitational Force allows us to estimate the mass of the Earth, as follows:

Mass of the Earth from gravitational acceleration
This is an example of one of the powerful implications of Newton's Law of Gravity: It gives us a way to use the motions of objects under the influence of their mutual gravitation to measure the masses of planets, stars, galaxies, etc.

The Orbit of the Moon

Falling apples are one thing, but what about the Moon?
What keeps the Moon in orbit around the Earth?

The Law of Inertia (Newton's First Law of Motion) predicts:

If there were no gravitational force acting between the Moon and the Earth, the Moon would travel in a straight line at a constant speed.
But, of course the Moon really moves along a curved path:
It is deflected from a straight-line path by the force of gravity.
This causes the Moon to fall a little bit towards the Earth at the same time at it moves to one side.

The Fall of the Moon

How far does the Moon fall around the Earth in 1 second? How far does an apple fall on the Earth during the first second? Newton also knew that:

Summarizing the numbers:

The Moon:

The Apple:

The ratio of the deflections of the Apple and the Moon in 1 second is ratio of their accelerations:

Putting all the info we have together, we get the following:

This predicts that the deflection of the Moon in 1 second necessary to keep it in orbit around the Earth should be 1/3600th the deflection of an apple during the first second of its fall to the Earth.

Observations vs. Prediction

Is this right?

Previously we found from observations that the deflections of the Moon and apple in 1 second are:

Gravity predicts that

The agreement is essentially perfect!

This demonstrates that the same law of gravity applies to both the apple and the Moon! Both feel the gravity of the Earth in the form of a force that gets weaker as the square of their distance from the center of the Earth.


So why does the Moon orbit the Earth?

If the Moon is falling a little towards the Earth, just like an apple dropped on the surface, why does the Moon travel around the Earth in an orbit instead of falling onto it?

The way to answer this question is to first consider what would happen if there was no gravity acting:

Question:
How far would the Moon travel in a straight line in 1 second if there were no gravity acting?

Answer:
About 1000 meters.

At the same time, the Moon's motion along this straight-line path would also cause it to move away from the Earth.

Question:
How far away from the Earth would the Moon move in 1 second if no gravity were acting?

Answer:
About 0.00136 meters!
In round numbers, the amount the Moon falls towards the Earth due to gravity is just enough to offset the straight-line path it would take if gravity were not acting to deflect it. This balance effectively closes the loop.

We have therefore reached a startling conclusion:

The Moon is really perpetually falling around the Earth!
This is a totally different way of looking at an "orbit" under the influence of gravity.

While at first sight the fall of an apple and the orbit of the Moon appear to be two completely different phenomena, viewed in light of Newton's laws of motion, they are in fact different manifestations of the same thing! The fall of the Moon around the Earth is the same kind of motion as the fall of an apple to the Earth. Both are described by the same three laws of motion, and both feel a gravitational force described by the same, universal force law.


Universal Gravitation

The Law of Gravity is Universal: and so on throughout the Universe.
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Updated: 2007 October 10
Copyright Richard W. Pogge, All Rights Reserved.