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WALTER LEWIN: If we take the shuttle as an example of a near earth orbit,
so we have the shuttle.
The shuttle may be 400 kilometers above the earth's surface.
So we have to add to the radius of the earth 400 kilometers.
So you end up with about 6,800 kilometers for the radius of the orbit
of the shuttle.
And you substitute that in here, the mass of the earth and the
gravitational constant.
You'll find that T is about 90 minutes.
It's about 1 and 1/2 hours.
The shuttle takes about 1 and 1/2 hours to go around.
And the speed, that tangential speed is very close to eight
kilometers per second.
And that holds for all near Earth orbit satellites.
Whether are they are 400, or 500, or 600 kilometers, that
doesn't change very much.
If you take the moon, the Moon is much further away than the shuttle.
And you take the distance to the Moon, which is some 385,000 kilometers.
You substitute that in this equation.
You will find that the period for the Moon to go around the Earth is about
27 and 1/2 days.
And its speed is only 1 kilometer per second.
It is much further out, R is much larger.
And so you see the speed will be much lower.
If you take the Earth itself around the Sun because we can
use all these equations.
Replace the mass of the earth by the mass of the Sun and then we can do
this for planets.
So if we take the Earth around the Sun.
Then we have to put in the mass of the Sun, which is about 2 times 10 to the
30 kilograms.
And the distance from Earth to the Sun, we've seen that before.
I call that the distance from the Sun to the Earth, is about 150 million
kilometers.
Forgive me for mixing up meters with kilometers, but you have to convert
that, of course, to meters.
And when you calculate how long it takes the Earth to go around the Sun,
no surprise.
You will find 365 and 1/2 days.
So the simplest substitution of these two quantities in the
equation that I have here.
And that I have here.
The velocity of the Earth in orbit is about 30 kilometers per second.
It's a substantial speed, by the way, that the Earth is
going around the Sun.
30 kilometers per second, way higher than the speed that the shuttle is in
around the orbit, around the earth, which is only eight
kilometers per second.
Jupiter is five times further away than the earth.
And so the time for Jupiter to go around goes with 5 to the
power of 1 and 1/2.
That's about 12, so it takes Jupiter about 12 years to go around the Sun.
Notice that this period is independent of the mass of the little satellite.
And that was very unfortunate for the Americans when on October 4, 1957,
Sputnik was launched.
They could find the radius very easily because they knew the period that it
took Sputnik to go around the earth.
It was about 96 minutes.
They could calculate the velocity.
They could calculate the radius, but they had no clue about the mass.
And that was a key piece of the ingredient that the Americans wanted
because if the mass was very large of Sputnik, that would indicate, of
course, that the Russians had very powerful rockets.
But you cannot tell the mass from the orbital parameters.
It's independent of mass.
Whether you have a very light object or very heavy satellite, they have the
same velocity in orbit if they're at the same distance, and they have the
same orbital period.
I mentioned earlier, notice that the orbital period and the escape velocity
vary by the square root of 2.
If you're at a particular position, for instance, you're at a particular
position around the Earth, here at a satellite.
If you want to escape from this, you will need a speed which is the square
root of 2 times larger than that orbital velocity.
And so if you wanted to escape from the Earth, then you need your 11.2
kilometers.
We have it there.
If you're in near Earth in orbit you are 8 kilometers per second.
And eight times the square root of 2 is exactly that 11.2.
So you see that connection is always through this square root of 2.