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I will now do a presentation
on the center of mass.

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And the center mass, hopefully,
is something that

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will be a little bit intuitive
to you, and it actually has

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some very neat applications.

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So in very simple terms, the
center of mass is a point.

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Let me draw an object.

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Let's say that this
is my object.

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Let's say it's a ruler.

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This ruler, it exists,
so it has some mass.

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And my question to you is
what is the center mass?

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And you say, Sal, well, in order
to know figure out the

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center mass, you have to tell me
what the center of mass is.

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And what I tell you is the
center mass is a point, and it

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actually doesn't have to even
be a point in the object.

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I'll do an example soon
where it won't be.

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But it's a point.

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And at that point, for dealing
with this object as a whole or

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the mass of the object as a
whole, we can pretend that the

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entire mass exists
at that point.

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And what do I mean
by saying that?

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Well, let's say that the
center of mass is here.

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And I'll tell you why
I picked this point.

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Because that is pretty close
to where the center

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of mass will be.

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If the center of mass is there,
and let's say the mass

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of this entire ruler is, I don't
know, 10 kilograms. This

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ruler, if a force is applied at
the center of mass, let's

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say 10 Newtons, so the mass
of the whole ruler is 10

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kilograms. If a force is applied
at the center of mass,

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this ruler will accelerate the
same exact way as would a

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point mass.

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Let's say that we just had a
little dot, but that little

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dot had the same mass, 10
kilograms, and we were to push

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on that dot with 10 Newtons.

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In either case, in the case of
the ruler, we would accelerate

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upwards at what?

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Force divided by mass, so we
would accelerate upwards at 1

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meter per second squared.

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And in this case of this
point mass, we would

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accelerate that point.

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When I say point mass, I'm just
saying something really,

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really small, but it has a mass
of 10 kilograms, so it's

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much smaller, but it has the
same mass as this ruler.

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This would also accelerate
upwards with a magnitude of 1

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meters per second squared.

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So why is this useful to us?

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Well, sometimes we have some
really crazy objects and we

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want to figure out exactly
what it does.

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If we know its center of mass
first, we can know how that

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object will behave without
having to worry about the

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shape of that object.

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And I'll give you a really easy
way of realizing where

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the center of mass is.

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If the object has a uniform
distribution-- when I say

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that, it means, for simple
purposes, if it's made out of

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the same thing and that thing
that it's made out of, its

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density, doesn't really change
throughout the object, the

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center of mass will be the
object's geometric center.

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So in this case, this
ruler's almost a

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one-dimensional object.

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We just went halfway.

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The distance from here to here
and the distance from here to

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here are the same.

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This is the center of mass.

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If we had a two-dimensional
object, let's say we had this

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triangle and we want to figure
out its center of mass, it'll

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be the center in
two dimensions.

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So it'll be something
like that.

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Now, if I had another situation,
let's say I have

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this square.

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I don't know if that's big
enough for you to see.

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I need to draw it a
little thicker.

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Let's say I have this square,
but let's say that half of

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this square is made from lead.

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And let's say the other half
of the square is made from

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something lighter than lead.

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It's made of styrofoam.

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That is lighter than lead.

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So in this situation, the center
of mass isn't going to

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be the geographic center.

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I don't know how much denser
lead is than styrofoam, but

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the center of mass is going to
be someplace closer to the

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right because this object does
not have a uniform density.

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It'll actually depend on how
much denser the lead is than

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the styrofoam, which
I don't know.

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But hopefully, that gives you a
little intuition of what the

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center of mass is.

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And now I'll tell you something
a little more

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interesting.

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Every problem we have done so
far, we actually made the

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simplifying assumption that
the force acts on

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the center of mass.

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So if I have an object, let's
say the object that

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looks like a horse.

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Let's say that object.

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If this is the object's center
of mass, I don't know where

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the horse's center of mass
normally is, but let's say a

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horse's center of
mass is here.

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If I apply a force directly on
that center of mass, then the

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object will move in the
direction of that force with

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the appropriate acceleration.

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We could divide the force by the
mass of the entire horse

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and we would figure out the

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acceleration in that direction.

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But now I will throw in a twist.
And actually, every

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problem we did, all of these
Newton's Law's problems, we

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assumed that the force acted
at the center of mass.

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But something more interesting
happens if the force acts away

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from the center of mass.

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Let me actually take
that ruler example.

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I don't know why I even
drew the horse.

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If I have this ruler again and
this is the center of mass, as

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we said, any force that we act
on the center of mass, the

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whole object will move in the
direction of the force.

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It'll be shifted by the
force, essentially.

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Now, this is what's
interesting.

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If that's the center of mass and
if I were to apply a force

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someplace else away from the
center of mass, let' say I

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apply a force here, I want you
to think about for a second

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what will probably happen
to the object.

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Well, it turns out that the
object will rotate.

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And so think about if we're on
the space shuttle or we're in

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deep space or something, and
if I have a ruler, and if I

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just push at one end of the
ruler, what's going to happen?

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Am I just going to push the
whole ruler or is the whole

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ruler is going to rotate?

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And hopefully, your intuition
is correct.

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The whole ruler will rotate
around the center of mass.

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And in general, if you were to
throw a monkey wrench at

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someone, and I don't recommend
that you do, but if you did,

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and while the monkey wrench is
spinning in the air, it's

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spinning around its
center of mass.

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Same for a knife.

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If you're a knife catcher,
that's something you should

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think about, that the object,
when it's free, when it's not

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fixed to any point, it rotates
around its center of mass, and

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that's very interesting.

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So you can actually throw random
objects, and that point

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at which it rotates
around, that's the

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object's center of mass.

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That's an experiment that you
should do in an open field

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around no one else.

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Now, with all of this, and
I'll actually in the next

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video tell you what this is.

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When you have a force that
causes rotational motion as

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opposed to a shifting motion,
that's torque, but we'll do

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that in the next video.

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But now I'll show you just a
cool example of how the center

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of mass is relevant in everyday
applications, like

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high jumping.

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So in general, let's say
that this is a bar.

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This is a side view of a
bar, and this is the

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thing holding the bar.

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And a guy wants to jump
over the bar.

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His center of mass is-- most
people's center of mass is

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around their gut.

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I think evolutionarily that's
why our gut is there, because

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it's close to our
center of mass.

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So there's two ways to jump.

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You could just jump straight
over the bar, like a hurdle

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jump, in which case your center
of mass would have to

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cross over the bar.

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And we could figure out this
mass, and we can figure out

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how much energy and how much
force is required to propel a

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mass that high because we know
projectile motion and we know

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all of Newton's laws.

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But what you see a lot in the
Olympics is people doing a

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very strange type of jump,
where, when they're going over

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the bar, they look something
like this.

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Their backs are arched
over the bar.

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Not a good picture.

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But what happens when someone
arches their back over

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the bar like this?

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I hope you get the point.

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This is the bar right here.

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Well, it's interesting.

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If you took the average of
this person's density and

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figured out his geometric center
and all of that, the

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center of mass in this
situation, if someone jumps

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like that, actually travels
below the bar.

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Because the person arches their
back so much, if you

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took the average of the total
mass of where the person is,

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their center of mass actually
goes below the bar.

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And because of that, you can
clear a bar without having

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your center of mass go as high
as the bar and so you need

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less force to do it.

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Or another way to say it, with
the same force, you could

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clear a higher bar.

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,

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Hopefully, I didn't confuse you,
but that's exactly why

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these high jumpers arch their
back, so that their center of

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mass is actually below the bar
and they don't have to exert

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as much force.

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Anyway, hopefully you found that
to be a vaguely useful

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introduction to the center of
mass, and I'll see you in the

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next video on torque.

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