An Introduction to Solar System Astronomy
Prof. Richard Pogge, MTWThF 9:30
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]
Explored the rate of falling bodies by dropping different weights, or sliding them down inclined planes.
Law of Falling Bodies:
Gravity is an Attractive force:
Gravity is a Universal force:
Gravity is a Mutual force:
The masses of the two objects:
The distance between them:
It does not depend on the shapes, colors, or compositions of the objects.
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"
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 experimentally.
What is the force of the Earth on the apple?
F = GME Mapple/RE2
What is the apple's acceleration (2nd Law):
aapple = F/Mapple = GME/RE2 = 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.
What force does the the apple apply in return upon the Earth?
F = GME MA/RE2
How much does the Earth accelerate towards the apple?
a = F/ME = GMA/RE2
= 9.8 m/sec2 x (MA/ME) = very small amount (about 10-25 meters/sec2)
a = 9.8 meters/sec2
We can also measure the radius of the Earth using geometry (Eratosthenes):
RE=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:
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 Law of Inertia (Newton's 1st law) 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:
Call this quantity xmoon, the deflection of the orbiting moon in 1 second.
How far does an apple fall on the Earth during the first second?
Call this quantity xapple, the deflection of a falling apple in its 1 second of motion.
Newton also knew that:
Summarizing the numbers:
The ratio of the deflections of the Apple and the Moon in 1 second will be proportional to the ratio of how much they have been accelerated during that 1 second.
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.
Is this right?
Recalling that from before, we found 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.
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.