And if we solve this for velocity,
put it here under the equation,
we got the GmM / r squared = 2 pi r / p squared m / r.
See that the masses are going to cancel out the mass of the actual object itself,
the mass of the planet.
In this case, it doesn't matter, it's only the mass of the central object.
That's only true as long as the central object is much more massive.
But that's the case in all these cases here.
And solving through these,
we get that p squared GM = 4 pi squared r cubed.
Look at this, we got that p squared is proportional to r cube,
that's just what we had over here.
So this recovers Kepler's Law, but
it shows us also what those proportionality factors are.
Those proportionality factors are, well, there's just some numbers over here.
But G and M, M is the mass of that central object, the mass of the Sun.
If it's a planet, the mass of Jupiter, if it's a moon going around.
And so we can solve for the mass, the mass = 4 pi squared / G r cubed / p squared.
So all we have to do is figure out the radius of the orbit, period of the orbit,
and we get the mass of the thing in the middle.
So that point, Newton could go back and figure out the radius of those orbits
from observations of Galileo or the many observations after that.
Periods were very easy to determine.
And the mass, well, we're not quite there yet because there are two problems.
One is G, this is a gravitational constant, but
it was not well known at the time.
And actually, the other is R, we'll talk about that a minute.
First, lets talk about G, how do you measure what G is?
Newton simply said that the force was proportional to this
product of the mass divided by r squared and so that proportionality constant is G.
How do you measure G?
First, really measurement for G was in about 1797, by Cavendish.
And it's so famous, it's actually called the Cavendish experiment.
And it's a pretty simple idea, which is of course if any two masses attract
each other, if you can put two masses next to each other and
see the force that they exert towards each other, you've measured G.
And he did exactly this, he put a pair of
weights on a, it's a torsion spring.
Imagine like a long strip that is allowed to rotate one direction or
rotate the other direction.
And then he would take larger weights on either the front and
back, or he would switch their positions front and back and
watch this thing ever so slightly deflect.
And he could calibrate how much it took to deflect that.
And he measured the value for G,
that's something within 1% of the value that we know, today.
He actually did it, he didn't think of himself as measuring G.
He thought of himself as measuring the density of the Earth.
Nobody really knew what the density of the Earth was, but we knew what G,
the gravitational constant was, something like 10 meters per second squared.
And we knew how big the Earth was and
so, Cavendish's question was, what is the density of the Earth?
How much mass does it take to give this
amount of gravitational pull for the Earth?
And the only way to know that was to know how mass and gravity related,
which is to know G.
But at the time, Cavendish didn't think of it as G.
But in the end, if you take his density measurement inverted for G,
he had a very precise measurement of G.
So now, we can go measure the density of Jupiter, right?
