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## Newton's law of gravitation

# Acceleration due to gravity at the space station

AP.PHYS:

FLD‑2.B (EU)

, FLD‑2.B.1 (EK)

, FLD‑2.B.1.1 (LO)

, FLD‑2.B.2 (EK)

, FLD‑2.B.2.1 (LO)

## Video transcript

Most physics books will tell
you that the acceleration due to gravity near the
surface of the Earth is 9.81 meters per
second squared. And this is an approximation. And what I want to
do in this video is figure out if this is the
value we get when we actually use Newton's law of
universal gravitation. And that tells us that the
force of gravity between two objects-- and let's just
talk about the magnitude of the force of gravity
between two objects-- is equal to the universal
gravitational constant times the mass of one of the
bodies, M1, times the mass of the second body divided by
the distance between the center of masses of the bodies squared. So let's use this, the
universal law of gravitation to figure out what the
acceleration due to gravity should be at the
surface of the Earth. And I have a g right over here. I have the mass of the Earth,
which I've looked up over here. And we also have the
radius of the Earth. And for the sake of
this, we're going to assume that the distance
between the body, if we're at the the surface of the
Earth, the distance between that and the center of
the Earth is just going to be the
radius of the Earth. And so this will give us
the magnitude of the force. If we want to figure out the
magnitude of the acceleration, which this really is-- I
actually didn't write this is a vector. So this is just the magnitude
of the acceleration. If you wanted the acceleration,
which is a vector, you'd have to say downwards or
towards the center of the Earth in this case. But if you want
the acceleration, we just have to
remember that force is equal to mass
times acceleration. And if you wanted to
solve for acceleration you just divide both
sides times mass. So force divided by mass
is equal to acceleration. Or if you take the
magnitude of your force and you divide by
mass, you're going to get the magnitude
of your acceleration. This is a scalar quantity. This is a scalar
quantity right over here. So if you want the acceleration
due to gravity, you divide. Let's write this in terms of
the force of gravity on Earth. So the magnitude of
the force of gravity on Earth, this one
right over here. So this will be in
the case of Earth. I just wrote Earth
really, really small. So one of these masses
is going to be Earth. It's going to be this
mass right over here. And so if you wanted
the acceleration due to gravity at the
surface of the Earth, you would just have to
divide by the mass that is being accelerated
due to that force. And in this case, it
is the other mass. It is the mass that's
sitting on the surface. So let's divide both
sides by that mass. Let's divide both
sides by that mass. And this will give
us the magnitude of the acceleration on
that mass due to gravity. So this is equal to the
magnitude of acceleration, due to gravity. And the whole reason why this
is actually a simplifying thing is that these two, this M2
right over here and this M2 cancels out. And so the magnitude
of our acceleration due to gravity using Newton's
universal law of gravitation is just going to be this
expression right over here. It's going to be the
gravitational constant times the mass of the Earth
divided by the distance between the object's
center of mass and the center of the
mass of the Earth. And we're going to
assume that the object is right at the surface,
that its center of mass is right at the surface. So this is actually going to
be the radius of the Earth squared, so divided
by radius squared. Sometimes this is also viewed
as the gravitational field at the surface of the Earth. Because if you
multiply it by a mass, it tells you how much force
is pulling on that mass. But with that out of
the way, let's actually use a calculator to
calculate what this value is. And then what I want to do
is figure out, well, one, I want to compare
it to the value that the textbooks
give us and see, maybe, why it may or may
not be different. And then think
about how it changes as we get further
and further away from the surface of the Earth. And in particular, if
we get to an altitude that the space shuttle or the
International Space Station might be at, and this is at
an altitude of 400 kilometers is where it tends to
hang out, give or take a little bit, depending
on what it is up to. So first, let's just
figure out what this value is when we use a universal
law of gravitation. So let's get my calculator out. So we know what g is. It is 6.6738 times 10
to the negative 11. This EE button means, literally,
times 10 to the negative 11. So this is 6.6738 times
10 to the negative 11. And then I want to
multiply that times the mass of Earth, which
is right over here. That is 5.9722 times
10 to the 24th. So times 10 to the 24th power. And we want to divide that by
the radius of Earth squared. So divided by the
radius of Earth is-- so this is in kilometers. And I just want to make
sure that everything is the same units. So 6,371 kilometers--
actually, let me scroll over. Well, you can't see the
kilometers right now. But this is kilometers. It is the same thing
as 6,371,000 meters. If you just multiply
this by 1,000. Or you could even
write this as 6.371. 6.371 times 10 to
the sixth meters. And we're going to square this. That's the radius of the Earth. The distance between the center
of mass of Earth and the center of mass of this object,
which is sitting at the surface of the Earth. And so let's get our drum roll. And we get 9.8. And if we round, we actually
get something a little bit higher than what the
textbooks give us. We get 9.82. Let's just round. So we get 9.82-- 9.82
meters per second squared. And so you might say,
well, what's going on here? Why do we have this
discrepancy between what the universal law of
gravitation gives us and what the average
measured acceleration due to the force of gravity
at the surface of the Earth. And the discrepancy here, the
discrepancy between these two numbers, is really
because Earth is not a uniform sphere
of uniform density. And that's what we have
to assume over here when we use the universal
law of gravitation. It's actually a little bit
flatter than a perfect sphere. And it definitely does
not have uniform density. The different layers of the
Earth have different densities. You have all sorts of
different interactions. And then you also, if you
measure effective gravity, there's also a little bit of a
buoyancy effect from the air. Very, very, very,
very negligible, I don't know if it would have
been enough to change this. But there's other minor,
minor effects, irregularities. Earth is not a perfect sphere. It is not of uniform density. And that's what accounts
for the bulk of this. Now, with that out of the
way, what I'm curious about is what is the
acceleration due to gravity if we go up 400 kilometers? So now, the main difference
here, g will stay the same. The mass of Earth
will stay the same, but the radius is now
going to be different. Because now we're placing the
center of mass of our object-- whether it's a space station
or someone sitting in the space station, they're going to
be 400 kilometers higher. And I'm going to exaggerate
what 400 kilometers looks like. This is not drawn to scale. But now the radius is going
to be the radius of the Earth plus 400 kilometers. So now, for the case
of the space station, r is going to be not
6,371 kilometers. It's going to be 6,000--
we're going to add 400 to this-- 6,771
kilometers, which is the same thing as
6,771,000 meters, which is the same thing as 6.771
times 10 to the sixth meters. This is-- 1, 2, 3, 4, 5,
6-- 10 to the sixth meters. So let's go back
to our calculator. So second entry, that's
the last entry we had. And instead of 6.371
times 10 to the sixth, let's add 400
kilometers to that. So then we get 6.7. So we're adding 400 kilometers. So it was 371. Now it's 771 times
10 to the sixth. And what do we get? We get 8.69 meters
per second squared. So now the acceleration here is
8.69 meters per second squared. And you can verify that
the units work out. Because over here,
gravity is in meters cubed per kilogram
second squared. You multiply that times
the mass of the Earth, which is in kilograms. The kilograms cancel out
with these kilograms. And then you're dividing
by meters squared. So you divide this
by meters squared. You're left with meters
per second squared. So the units work out as well. So there's an important
thing to realize. And this is a misconception. We do a whole video
on it earlier, when we talk about the
universal law of gravitation, is that there is gravity when
you are in orbit up here. The only reason why it feels
like there's not gravity or it looks like
there's not gravity is that this space
station is moving so fast that it's
essentially in free fall. But it's moving so fast that
it keeps missing the Earth. And in the next video,
we'll figure out how fast does it have to
travel in order for it to stay in orbit, in order for it to not
plummet to Earth due to this, due to the force of gravity,
due to the acceleration that is occurring, this centripetal,
this center-seeking acceleration?