WEBVTT
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This is Giancoli Answers with Mr. Dychko.
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The distance from the center of the Earth
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to the spacecraft, we'll label it
*d* subscript *E*
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and then we have the distance to the moon
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and we know that this position,
where there is zero net force
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of gravity on the spacecraft,
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has to be closer to the moon
than it is to the Earth
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because the moon is so much less massive;
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we have to get closer to it for its force to
get up to the point where it can match
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the force exerted by this mass of Earth.
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Now, to reduce the number of variables,
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we can use this
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value for the distance between the
Earth and moon center's,
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384 times 10 to the 6 meters
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to express this *d* subscript *M*,
in terms of *d E*.
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So the distance to the moon
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is the total distance, which
we know, minus *d E*.
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And then when we read our
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gravity due to the Earth equals
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gravity due to the moon formula,
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we can substitute for *d M*
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and write *d* minus *d E* in its place.
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And then we have an equation with only
one unknown, which we can solve.
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So this is gravitational constant times
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mass of the Earth times
mass of the spacecraft
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divided by distance to the spacecraft
from the Earth squared
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equals the force of gravity exerted
by the moon,
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*G* mass of the moon times
mass of the spacecraft
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divided by the distance
to the moon squared.
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And the *G*'s cancel and so does
the mass of the spacecraft
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and so this position is
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does not depend on the spacecraft's
mass, as it turns out.
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And, we have also made a
substitution here for that
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distance to the moon in terms of
the distance from the Earth.
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So, we'll take the reciprocal
of both sides by
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raising both sides to the
exponent negative 1
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to get our unknown in the
numerator which is a
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easier place to work on it, with
unknowns in the numerator.
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And then we'll take the square
root of both sides
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to get rid of that square there.
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And so we have *d E* over square root of
mass of the Earth equals
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*d* minus *d E* divided by square root
mass of the moon.
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And then, we get rid of the denominators
here by multiplying
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by square root mass of the moon
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and also multiplying by square root
mass of the Earth both sides
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and mass of the Earth cancels there
leaving us with distance to the Earth
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times square root mass of the moon equals
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*d* times mass of the Earth
minus *d* to the Earth
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times square root mass of the Earth.
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And we'll put both the *d E* terms
on the same side
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and when we move this to the left,
it becomes positive
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and so we can factor out
this common factor,
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distance to the Earth multiplied by
square root mass of the moon
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plus square root mass of the Earth
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equals total separation between
the Earth and the moon
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times square root mass of the Earth
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and then divide both sides
by this bracket here
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to get that the distance to the Earth
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from this position where there's
zero force of gravity
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is separation between the
Earth and the moon
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times square root mass of
the Earth divided by
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the sum of the square roots of the masses
of the Earth and the moon.
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And on the calculator, it looks like this.
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So, we have 384 times 10 to the 6 meters
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times square root 5.98 times 10 to the
24 kilograms—mass of the Earth—
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divided by square root of 7.35 times 10 to
the 22 kilograms—mass of the moon—
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plus square root mass of the Earth
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and we get 3.46 times 10 to the 8 meters
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is the distance from the Earth,
Earth's center I should say,
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to the point where there is zero net
force due to gravity.