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## Newton's laws and equilibrium

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# Center of mass

## Video transcript

I will now do a presentation
on the center of mass. And the center mass, hopefully,
is something that will be a little bit intuitive
to you, and it actually has some very neat applications. So in very simple terms, the
center of mass is a point. Let me draw an object. Let's say that this
is my object. Let's say it's a ruler. This ruler, it exists,
so it has some mass. And my question to you is
what is the center mass? And you say, Sal, well, in order
to know figure out the center mass, you have to tell me
what the center of mass is. And what I tell you is the
center mass is a point, and it actually doesn't have to even
be a point in the object. I'll do an example soon
where it won't be. But it's a point. And at that point, for dealing
with this object as a whole or the mass of the object as a
whole, we can pretend that the entire mass exists
at that point. And what do I mean
by saying that? Well, let's say that the
center of mass is here. And I'll tell you why
I picked this point. Because that is pretty close
to where the center of mass will be. If the center of mass is there,
and let's say the mass of this entire ruler is, I don't
know, 10 kilograms. This ruler, if a force is applied at
the center of mass, let's say 10 Newtons, so the mass
of the whole ruler is 10 kilograms. If a force is applied
at the center of mass, this ruler will accelerate the
same exact way as would a point mass. Let's say that we just had a
little dot, but that little dot had the same mass, 10
kilograms, and we were to push on that dot with 10 Newtons. In either case, in the case of
the ruler, we would accelerate upwards at what? Force divided by mass, so we
would accelerate upwards at 1 meter per second squared. And in this case of this
point mass, we would accelerate that point. When I say point mass, I'm just
saying something really, really small, but it has a mass
of 10 kilograms, so it's much smaller, but it has the
same mass as this ruler. This would also accelerate
upwards with a magnitude of 1 meters per second squared. So why is this useful to us? Well, sometimes we have some
really crazy objects and we want to figure out exactly
what it does. If we know its center of mass
first, we can know how that object will behave without
having to worry about the shape of that object. And I'll give you a really easy
way of realizing where the center of mass is. If the object has a uniform
distribution-- when I say that, it means, for simple
purposes, if it's made out of the same thing and that thing
that it's made out of, its density, doesn't really change
throughout the object, the center of mass will be the
object's geometric center. So in this case, this
ruler's almost a one-dimensional object. We just went halfway. The distance from here to here
and the distance from here to here are the same. This is the center of mass. If we had a two-dimensional
object, let's say we had this triangle and we want to figure
out its center of mass, it'll be the center in
two dimensions. So it'll be something
like that. Now, if I had another situation,
let's say I have this square. I don't know if that's big
enough for you to see. I need to draw it a
little thicker. Let's say I have this square,
but let's say that half of this square is made from lead. And let's say the other half
of the square is made from something lighter than lead. It's made of styrofoam. That is lighter than lead. So in this situation, the center
of mass isn't going to be the geographic center. I don't know how much denser
lead is than styrofoam, but the center of mass is going to
be someplace closer to the right because this object does
not have a uniform density. It'll actually depend on how
much denser the lead is than the styrofoam, which
I don't know. But hopefully, that gives you a
little intuition of what the center of mass is. And now I'll tell you something
a little more interesting. Every problem we have done so
far, we actually made the simplifying assumption that
the force acts on the center of mass. So if I have an object, let's
say the object that looks like a horse. Let's say that object. If this is the object's center
of mass, I don't know where the horse's center of mass
normally is, but let's say a horse's center of
mass is here. If I apply a force directly on
that center of mass, then the object will move in the
direction of that force with the appropriate acceleration. We could divide the force by the
mass of the entire horse and we would figure out the acceleration in that direction. But now I will throw in a twist.
And actually, every problem we did, all of these
Newton's Law's problems, we assumed that the force acted
at the center of mass. But something more interesting
happens if the force acts away from the center of mass. Let me actually take
that ruler example. I don't know why I even
drew the horse. If I have this ruler again and
this is the center of mass, as we said, any force that we act
on the center of mass, the whole object will move in the
direction of the force. It'll be shifted by the
force, essentially. Now, this is what's
interesting. If that's the center of mass and
if I were to apply a force someplace else away from the
center of mass, let' say I apply a force here, I want you
to think about for a second what will probably happen
to the object. Well, it turns out that the
object will rotate. And so think about if we're on
the space shuttle or we're in deep space or something, and
if I have a ruler, and if I just push at one end of the
ruler, what's going to happen? Am I just going to push the
whole ruler or is the whole ruler is going to rotate? And hopefully, your intuition
is correct. The whole ruler will rotate
around the center of mass. And in general, if you were to
throw a monkey wrench at someone, and I don't recommend
that you do, but if you did, and while the monkey wrench is
spinning in the air, it's spinning around its
center of mass. Same for a knife. If you're a knife catcher,
that's something you should think about, that the object,
when it's free, when it's not fixed to any point, it rotates
around its center of mass, and that's very interesting. So you can actually throw random
objects, and that point at which it rotates
around, that's the object's center of mass. That's an experiment that you
should do in an open field around no one else. Now, with all of this, and
I'll actually in the next video tell you what this is. When you have a force that
causes rotational motion as opposed to a shifting motion,
that's torque, but we'll do that in the next video. But now I'll show you just a
cool example of how the center of mass is relevant in everyday
applications, like high jumping. So in general, let's say
that this is a bar. This is a side view of a
bar, and this is the thing holding the bar. And a guy wants to jump
over the bar. His center of mass is-- most
people's center of mass is around their gut. I think evolutionarily that's
why our gut is there, because it's close to our
center of mass. So there's two ways to jump. You could just jump straight
over the bar, like a hurdle jump, in which case your center
of mass would have to cross over the bar. And we could figure out this
mass, and we can figure out how much energy and how much
force is required to propel a mass that high because we know
projectile motion and we know all of Newton's laws. But what you see a lot in the
Olympics is people doing a very strange type of jump,
where, when they're going over the bar, they look something
like this. Their backs are arched
over the bar. Not a good picture. But what happens when someone
arches their back over the bar like this? I hope you get the point. This is the bar right here. Well, it's interesting. If you took the average of
this person's density and figured out his geometric center
and all of that, the center of mass in this
situation, if someone jumps like that, actually travels
below the bar. Because the person arches their
back so much, if you took the average of the total
mass of where the person is, their center of mass actually
goes below the bar. And because of that, you can
clear a bar without having your center of mass go as high
as the bar and so you need less force to do it. Or another way to say it, with
the same force, you could clear a higher bar. , Hopefully, I didn't confuse you,
but that's exactly why these high jumpers arch their
back, so that their center of mass is actually below the bar
and they don't have to exert as much force. Anyway, hopefully you found that
to be a vaguely useful introduction to the center of
mass, and I'll see you in the next video on torque.