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## Kinematic formulas and projectile motion

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# Viewing g as the value of Earth's gravitational field near the surface

## Video transcript

What I want to do in this
video is think about the two different ways of
interpreting lowercase g. Which as we've talked about
before, many textbooks will give you as either 9.81
meters per second squared downward or towards
the Earth's center. Or sometimes it's given with
a negative quantity that signifies the direction, which
is essentially downwards, negative 9.81 meters
per second squared. And probably the
most typical way to interpret this value, as
the acceleration due to gravity near Earth's surface for
an object in free fall. And this is what we're going
to focus on this video. And the reason why I'm
stressing this last part is because we know
of many objects that are near the surface
of the Earth that are not in free fall. For example, I am near the
surface of the Earth right now, and I am not in free fall. What's happening to me right
now is I'm sitting in a chair. And so this is my chair--
draw a little stick drawing on my chair,
and this is me. And let's just
say that the chair is supporting all my weight. So I have-- my legs
are flying in the air. So this is me. And so what's
happening right now? If I were in free fall,
I would be accelerating towards the center of
the Earth at 9.81 meters per second squared. But what's happening is, all
of the force due to gravity is being completely
offset by the normal force from the surface of the
chair onto my pants, and so this is normal force. And now I'll make
them both as vectors. So the net force in my
situation-- the net force is equal to 0, especially
in this vertical direction. And because the net
force is equal to 0, I am not accelerating towards
the center of the Earth. I am not in free fall. And because this 9.81
meters per second squared still seems relevant
to my situation-- I'll talk about
that in a second. But I'm not an
object in free fall. Another way to interpret this
is not as the acceleration due to gravity near
Earth's surface for an object in free
fall, although it is that-- a maybe more
general way to interpret this is the gravitational-- or
Earth's gravitational field. Or it's really the average
acceleration, or the average, because it actually
changes slightly throughout the
surface of the Earth. But another way to view this, as
the average gravitational field at Earth's surface. Let me write it
that way in pink. So the average gravitational
field-- and we'll talk about what a field
means in the physics context in a second-- the
average gravitational field at Earth's surface. And this is a little bit
more of an abstract thing-- we'll talk about
that in a second-- but it does help
us think about how g is related to
this scenario where I am not an object in free fall. A field, when you think of
it in the physics context-- slightly more
abstract notion when you start thinking about it
in the mathematics context-- but in the physics
context, a field is just something that
associates a quantity with every point in space. So this is just a quantity
with every point in space. And it can actually be a
scalar quantity, in which case we call it a scalar
field, and in which case it would just be a value. Or it could be a
vector quantity, which would be a magnitude
and a direction associated with every point in space. In which case you are
dealing with a vector field. And the reason why
this is called a field is, because at near
Earth's surface, if you give me a mass--
so for example-- actually, I don't know what my
mass is in kilograms. But if you're near
Earth's surface and you give me a
mass-- so let's say that mass right over
there is 10 kilograms-- you can use g to figure out
the actual force of gravity on that object at
that point in space. So for example, if this
has a mass of 10 kilograms, then we know-- and
this right over here is the surface of the Earth, so
that's the center of the Earth. So it actually associates
a vector quantity whose magnitude,--
so its direction is towards the center of the
Earth, and the magnitude of this vector quantity is
going to be the mass times g. And you could take--
since we're already specifying the
direction, we could say 9.81 meters
per second squared towards the center of the Earth. And so in this
situation, it would be 10 kilograms times 9.81
meters per second squared. Which is 98.1. And even this I've
rounded a little bit, so it's actually
approximate number. 98.1 kilogram meters
per second squared which is the unit of
force, or 98.1 newtons. And this thing might
not be in free fall, so this is why g
is relevant even in a situation where the
object isn't in free fall. g has given us the
force per unit mass-- the force per mass of
gravity on an object near the surface of the Earth. Another way to think
about it-- so this is the average gravitational
field, and what it's giving is force per mass. So you give me a mass
near Earth's surface-- whether it's an
object in free fall or not-- you multiply
that mass times g, because it's giving
you force per mass, and it will give you the
force of gravity acting on that object near the
surface of the Earth, whether or not
it's in free fall. So I just want to make
this little distinction, because although g
tends to be referred to this way right over here. Sometimes you might
encounter a stickler who says oh no, no, no, no,
no but g is relevant even when an object is not in free fall. You obviously can't say
that my acceleration when I'm sitting in my chair is
9.81 meters per second squared towards the center of the Earth. I am not accelerating towards
the center of the Earth. And so they'll say,
oh no, no, no, no, you can't just call
this acceleration. It is true, it is
the acceleration when an object is in free fall
near the surface of the Earth-- if you don't have really air
resistance, if the net force really is the force of
gravity-- then this really would be the object's
acceleration. But it becomes
relevant, and we know most objects that we know
of aren't in free fall. Obviously, an object in free
fall doesn't stay in free fall for long. It eventually hits something. But we know that
now g is actually relevant to all objects. It tells us the force
per mass And it's tempting to call it
always acceleration-- because the units
are acceleration-- but even when you
talk about in terms of the gravitational field,
it's still the same quantity. It still has the exact same
units, the same magnitude, and the same
direction-- it's just a different way of viewing it. Here, acceleration for
an object in free fall. Here, something to
multiply by mass to figure out the
force due to gravity.