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.