WEBVTT
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The little prince's
asteroid B612
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is being orbited
by a small body.
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It has a mass m.
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And it goes around this asteroid
B612 that has a mass m1.
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We should add a
coordinate system.
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We always know about r-hat
goes radially outward.
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And we know from the
universal law of gravitation
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that we have mutual attraction
between these bodies.
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And actually, that's
going to point inward.
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And because we're having
orbital motion here,
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so circular motion,
the acceleration,
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the radial component
of the acceleration
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will also point inward.
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We need to consider all of that
for our f equals ma analysis
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that we are going to do now,
because the little prince wants
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to know how far this little
body is away from his asteroid.
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So we have the gravitational
universal law here,
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minus Gmm1 over the
distance squared.
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So the distance here
between the two planets,
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r, which is what we
want to calculate.
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And then we have over
here for circular motion,
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the description of
mr omega squared.
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Now, the little prince
can't measure omega.
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But the little prince
has a little time clock.
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So what he can measure
is the period from here
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until he sees the body again.
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And that is 2pi over omega.
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So we can add that in here.
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mr 4pi squared over T squared.
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And this m here will cancel out.
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And we have to solve this for r.
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What we're going to see--
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Oh and of course,
we have a minus sign
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here because in the life
of the little prince,
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of course gravitational
acceleration
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is not going outwards.
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It's going inward, so we better
give this a minus here and here
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as well.
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And we will actually
see that that then
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cancels out against this one.
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And we're going to
solve this for r.
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So we get r cubed, actually.
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And then we have Gm1
over 4pi squared.
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And here, we have T squared.
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And you can also just write that
as Gm1 or pi squared T squared.
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And then we have third root.
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So you might have seen
this equation here.
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This is actually Kepler's law.
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It describes the motion of
the planets around the sun.
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Well, it really only
does it if the motions
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are fairly circular.
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For elliptical orbits, it is
not such a good approximation,
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although Kepler derived it
like that quite a while ago.
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And that was really
an astonishing result.
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So here, we have this
again that the cube
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of the distance
between two objects
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is proportional to the square
of the period of the orbiting
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time.