So Newton has this beginnings of an 
understanding of the science of 
mechanics, and, the first problem to 
which he's going to apply it, is Kepler's 
recent results about the motion of the 
planets around the sun, and Newton has, a 
beautiful chain of logic that we can 
actually follow, that allows him to 
derive the fundamental physical principle 
that we suggest that might be underlying 
Kepler's answer. 
Newton's mechanical, understanding tells 
him that as we saw, if an object of mass 
m is moving around in a circle of radius 
R with uniform speed v, which is a 
simplified version of the elliptical 
motion that Kepler actually studied, 
Newton was doing the general case. 
We know that moving around in a circle 
involves a centripetal acceleration 
towards the center of the circle. 
So if you see anywhere in the universe an 
object that is governed by Newton's laws, 
like everything is, moving around in a 
circle of radius R with speed V then 
there is a force acting on that object 
and attracting it towards the center of 
the circle in which it is moving. 
And the magnitude of that force is the 
centripetal acceleration that we 
computed, v^2 / R times the mass of the 
object because F = ma. 
If you know the acceleration you can 
deduce the force. 
Now Newton follows. 
The moon orbits the earth. 
The moon is actually moving in a circle 
about the earth. 
So we know that there is a force acting 
on the moon, which we can compute if we 
know the radius of the moon's orbit. 
And thence because you know how long it 
takes to go around you know its velocity. 
If you knew the mass of the moon you 
could compute the force that the earth 
applies to the moon in order to keep it 
moving in a circle. 
Why do I think the earth applies it? 
Well because it's directed towards the 
earth, that's a natural guess. 
But there's another ingredient here. 
We know that the earth attracts 
everything. 
Little rubber balls fall onto the earth 
because there's this attractive force we 
call gravity. 
Apples fall on people's heads 
apocryphally. 
Could it be that the force that keeps the 
moon in orbit around the earth is the 
same as the one that makes tennis balls 
fall? 
Well Newton says, 
probably let's take this seriously. 
Extrapolating that, let's look at another 
system. 
we know Kepler found that the planets 
orbit the sun in, well almost, perfectly 
circular orbits at almost uniform speed. 
again, that means that acting on each 
planet, is a force directed towards the 
center of the orbit, in other words, the 
sun. 
And we can compute that force. 
So it makes sense to guess that the sun 
will be applying this force. 
And the sun, therefore, applies a force 
on all the planets. 
Uh-huh. 
Then by Newton's third law, it would 
follow that since the Sun must be 
applying force to all the planets, among 
them the Earth, the Earth also applies an 
equal and opposite force to the Sun. 
In other words, if the Sun applies a 
force to the Earth directly towards the 
Sun, it follows that the Earth applies a 
force to the Sun directly back towards 
the Earth. 
And now, we have lots of cases where the 
earth is applying a force to things 
directed towards the center of the earth. 
Perhaps that force too is the same force 
that makes the moon orbit the earth and 
apples fall on people's heads. 
It all hangs together in a beautiful 
framework and we can do the math. 
So let's start with the idea of something 
orbiting the Sun. 
So we'll have a planet, we'll call it 
mass mp, p stands for planet, any planet. 
And if it orbits the Sun at radius R with 
speed v then Newton's understanding of 
mechanics tells us that the force the Sun 
applies to that particular planet is the 
mass of the planet times v^2 / R for that 
planet. 
two clips ago when we studied Kepler's 
laws, we found, that in fact for any 
planet that orbits the sun, there is a 
relation between v and R, that is 
independent of which planet, you are 
considering. 
For all of the planets, the square of 
their speed v^2, is 4 pi squared divided 
by this interesting constant, that I 
called K for Kepler, times, the radius of 
the planet's orbit. 
So I can, put this expression into that, 
and I can find, F is equal to the mass of 
the planet times the square of its 
velocity. 
4 pi squared over KR. 
Divided by the radius of its orbit. 
Or rearranging things a moment for a bit 
four pi squared divided by K times the 
mass of the planet divided by R^2. 
Why is this a way to write it? 
Because the mass of the planet is a 
property of the planet. 
R is something that changes between 
different planets. 
4 Pi squared over K is the same for all 
planets. 
So, let's see what we do with this. 
Well, remember the idea was that it's all 
going to hang together it's all going to 
be the same force. 
The sun applies the force given by this 
expression we found to each planet and to 
all the planets it applies the same the, 
the, the force given by the same constant 
K. 
By Newton's third law, each planet 
applies a force of that magnitude, this 
precise magnitude, to the sun. 
Now universality says that the force that 
the planet applies to the sun, the force 
that the sun applies to the planet, 
should follow from some physical 
principle. 
It doesn't say well, 
the sun is one thing and the planet is 
another. 
They're both physical object, and so 
symmetry between the two objects tells 
us, that if it's proportional to the mass 
of the planet, it should be proportional 
to the mass of the sun, and of course I 
can, do that without any problem I can, 
say that there is some constant which I 
will call G, and I, if I write that 
constant such that it's related to K. 
And to the mass of the sun. 
By this relation. 
Then the force e, expression that I'm 
writing is equivalent to F is G times the 
mass of the Sun times the mass of the 
planet divided by R^2. 
The advantage of this formulation over 
this one is that here it's clear that 
there is a symmetry. 
What the planet does to the Sun, the Sun 
does to the planet. 
The force between them is equal in 
magnitude and opposite in direction. 
And so there's this symmetric way to 
write it. 
But it was clear already that this form, 
object here doesn't depend on the planet. 
So it has to be in, some property of the 
Sun. 
The only wild leap here is that the 
property of the sun it depends on is the 
same property. 
The mass and the dependence is the same 
as the property of the planet on which it 
depends since it determines the force in 
a symmetric way. 
And now, 
we can jump to the fact that it's much 
more universal. 
Remember, the sun attracts the planets. 
The planets attract the sun. 
Earth attracts the moon and the moon 
pulls on the Earth. 
And in fact, the only way to get a 
consistent, understanding of this is to 
imagine that the force of gravity is 
truly universal. 
Any object applies a force to any other 
object in the entire universe. 
And the force is proportional to the mass 
of two objects. 
It's inversely proportional to the square 
of the distance between them. 
And, there's this universal constant, 
that gravitation, that defines what 
gravitation is and how powerful it is, 
which very appropriately is given the 
name Newton's constant, the, First 
measurement we have of Newton's constant 
is in 1798 by Cavendish, who actually 
measures the gravitational attraction 
between two metal balls, whose masses he 
carefully measured separately. 
So he can actually measure the distance 
and the masses. 
And find the force, and extract a value 
for G. 
And G is in the units that are convenient 
to us, given by this number of 6.67 * 
10^-11 Newtons times meters squared per 
kilogram squared. 
So that when you multiply it by two 
masses, kilograms squared divided by 
meters, you get a force in Newtons. 
So we have this equation. 
And, the way in which it is superior to 
everything that we have seen so far 
cannot be overstated. 
This is a universal statement about 
anything attracting anything else. 
And from this follows, for example, 
Kepler's Third Law, and of course, by a 
little more math, all of his other laws, 
describing the motion of the planets, the 
motion of the Moon, and as we shall see, 
plenty more things. 
So we found the deep underlying physics 
that we were looking for. 
Newton found it, and we have gotten 
there. 
So let's think a little bit about orbits. 
the moon, as we said, has a force applied 
to it. 
The earth is attracting the moon 
gravitationally. 
And one might ask, why is does not the 
moon fall on earth? 
And the answer is that the moon is 
falling on earth. 
Remember moving in a circle is motion 
with a constant acceleration towards the 
earth. 
Just as my little rubber ball was 
accelerated down towards earth, so the 
moon is accelerated towards earth with a 
constant acceleration. 
To demonstrate this idea, we go back to 
our pink bowling ball. 
You'll notice that when I hold it away 
from me and let it go, the Earth's 
gravity will always make it fall towards 
my head since I'm standing under the 
suspension point, whatever direction I 
let it go in. 
But if I give it an initial velocity that 
is just right, it will, happily circle 
about my head in a roughly circular orbit 
while accelerating towards the center at 
all times. 
And we also see that by giving it either 
a larger or smaller initial velocity, I 
get elongated orbits. 
They're the analogue of elliptical orbits 
in the case of actually gravitational 
motion around the earth or the sun. 
We'll do more detailed calculations of 
all this later, but for the moment we 
understand that orbiting is just a way of 
falling without ever hitting the ground, 
to paraphrase Douglas Adams. 
What we saw is that uniform circular 
motion about the center of gravity is a 
solution to Newton's equations. 
And if you think back this would require 
that I launch the bowling ball with 
precisely the speed predicted by the 
Kepler relation and its distance from the 
center of gravity. 
What happens if the initial velocity is 
either too large or too small? 
Well as with the bowling ball you get at, 
elongated orbits, and you will not be 
shocked to learn that in the case of 
Newtonian gravity and Newton solved these 
cases the result you get is precisely the 
elliptical orbits with the, center of 
gravity at one focus that Kepler had 
observed for the motion of the planets 
around the sun. 
And so, you find that there are all these 
elliptical orbits, and in fact. 
There are also open orbits like this 
hyperbolic one we see here which 
describes the motion of an object that 
comes in from infinity and recedes off to 
infinite distance. 
Of course, if you start along any point 
of this orbit with the correct velocity 
then you will pre- reproduce the later 
part of the orbit. 
So these describe the motions of objects 
but are not bound to the gravitating blue 
ball over there but rather are deflected 
by it. 
We'll return to orbits like this in the 
future. 
So, 
this is very nice. 
But, it gets even more general than that. 
Newton's law is universal, so it'll apply 
to any two objects that are orbiting 
under the mutual gravity. 
A little more work shows that we've a 
little bit been cavalier, we sort of 
assumed that Sun was stationary and earth 
orbiting it or the earth stationary and 
moon orbiting it. 
You should have known that there's 
something wrong there. 
Because if the moon is accelerating 
towards earth due to the earth's 
gravitational attraction, well then the 
earth must be accelerating towards the 
moon due to the moon's gravitational 
attraction. 
Of course, the acceleration of the earth 
is less than that of the moon, because 
the earth is more massive. 
And, this is the sort of the situation 
here, a very massive object and a lighter 
object are orbiting under the influence 
of gravity. 
The cross that is fixed there is the 
center of mass of the system. 
It's distance from the center of each 
object is inversely proportional to the 
mass of the object. 
So, it's closer to the mass of the 
object, it's the point at which these 
things will balance And but they both 
orbit each other, eat the center of each 
object describes the circle so they're 
always on opposite sides of this circle. 
Taking into account this recoil you find 
that in fact the distance between the two 
the radius or if its an elliptical orbit 
the semi major axis still satisfies 
Kepler's law no matter what the object is 
as it has nothing to do with the sun. 
The correct statement it turns out when 
you take into account the motion of the 
heavy object, is that the count, the 
constant that relates the square of the 
period to the cube of the semi-major axis 
is in fact, 4 pi squared over G times the 
total mass of the system. 
The sum of the two masses, now in the 
case of the sun and the Earth or the 
Earth and the moon. 
One of them is so much more massive than 
the other that you can approximate ma- 
the earth sun mass as the sun and the 
earth moon mass as the earth. 
This is tantamount to making the 
approximation that the sun, respectively 
the earth, does not move and only the 
lighter object moves. 
In, in cases like binary stars or maybe 
when you are completing the orbit of 
Jupiter around the sun. 
When the masses are closer to each other, 
then you need to apply this more careful 
calculation. 
This will be a very useful result to us. 
Because it means that if we see any two 
objects orbiting each other in the 
universe, and we can somehow find the 
period and the semi-major axis, we can 
compute their mass and, if you look in 
your texts and you find the mass of 
planets and galaxies and stars, you might 
ask yourself, how did somebody compute 
the mass of a distant star or for that 
matter Jupiter? 
And the answer is, you find something 
orbiting it, and you apply Kepler's 
relation. 
And so this will be an extremely powerful 
tool as we go on. 
All this theory, let's do an example and 
let's consider the International Space 
Station. 
The International Space Station orbits at 
an altitude above the ground of about 370 
kilometers, adding that to the earth's 
radius, we find that it's the radius of 
its orbit about the center of the earth, 
is about 6,700 kilometers. 
And so we can find its period. 
Its period is given by our relation P^2 = 
K * R^3 and plugging in our value for 
pay, K, we find 4 Pi squared over GM 
earth times R^3. 
Notice that I've neglected the mass of 
the international space station relative 
to the mass of the earth which is a very 
fair approximation. 
And when we plug in the numbers we find 
that the period is 
taking the square root, you get this 
expression. 
Extracting the 2 Pi, plugging in all the 
numbers. 
Newton's constant the mass of the Earth 
and remembering to convert the radius of 
the Earth to meters to match the units in 
which we wrote Newton's constant. 
We find a period of 55 hundred seconds or 
about 91 minutes, the International Space 
Station and all lower satellites, I mean 
nothing was special to the, International 
Space Station. 
Anything that orbits at about the Earth's 
surface, orbits the Earth every 90 
minutes. 
When you see satellites crossing the 
heavens, they all appear to move without 
uniform angular velocity. 
This is the reason, Kepler is telling us 
that they all orbit once every hour and a 
half. 
I hope that you're as impressed as I am 
with what we've managed to achieve by 
following Newton. 
We have found the fundamental, 
universal laws that underlay the 
regularities of Kepler's laws. 
in Newton's time, 
what to help people appreciate this was 
the understanding by Edmund Haley that 
comets, these weird objects that would 
appear in the sky move along the 
celestial sphere in some random direction 
then disappear again. 
Were actually objects that were at highly 
elliptical, highly eccentric elliptical 
orbits about the sun, spent most of their 
time far from the sun and invisible and 
were visible to us when they came near. 
He used data that had been collected to 
predict the appearance in 1705 of a 
particular comet in the sky over London 
to within a day. 
And luckily he got it just right, people 
stepped out on the day that Haley's Comet 
as it has come to be called showed up and 
the, the validity and the importance of 
Newton's result was understood. 
here's a beautiful image of the passage 
of comet Haley in 1986, it has a 75 year 
period as it orbits the sun. 
Try to figure out its semi major axis. 
Gravity works, we'll come back and do 
some more details. 
We have some more things to learn about 
how much information we can get out of 
Newton's theory. 
We'll turn to that in the next clip.