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Welcome back.

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We'll now use a little bit of
what we've learned about work

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and energy and the conservation
of energy and

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apply it to simple machines.

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And we'll learn a little bit
about mechanical advantage.

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So I've drawn a simple
lever here.

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And you've probably
been exposed to

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simple levers before.

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They're really just kind
of like a seesaw.

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This place where the
lever pivots.

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This is called a fulcrum.

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Just really the pivot point.

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And you can kind of view this
as either a seesaw or a big

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plank of wood on top of a
triangle, which essentially is

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what I've drawn.

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So in this example, I have
the big plank of wood.

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At one end I have this 10
newton weight, and I've

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written 10 in there.

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And what we're going to figure
out is one, how much force--

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well, we could figure out
a couple of things.

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How much force do I have
to apply here to

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just keep this level?

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Because this weight's going
to be pushing downwards.

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So it would naturally
want this whole

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lever to rotate clockwise.

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So what I want to figure out is,
how much force do I have

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to apply to either keep the
lever level or to actually

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rotate this lever
counterclockwise?

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And when I rotate the lever

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counterclockwise, what's happening?

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I'm pushing down on this
left-hand side, and I'm

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lifting this 10 newton block.

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So let's do a little thought
experiment and see what

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happens after I rotate this
lever a little bit.

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So let's say, what I've drawn
here in mauve, that's our

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starting position.

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And in yellow, I'm going to draw
the finishing position.

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So the finishing position
is going to look

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something like this.

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I'll try my best to draw it.

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The finishing position is
something like this.

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And also, one thing I want to
figure out, that I wanted to

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write, is let's say that the
distance, that this distance

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right here, from where I'm
applying the force to the

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fulcrum, let's say that
that distance is 2.

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And from the fulcrum to the
weight that I'm lifting, that

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distance is 1.

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Let's just say that, just for
the sake of argument.

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Let's say it's 2 meters and 1
meter, although it could be 2

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kilometers and 1 kilometer,
we'll soon see.

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And what I did is I pressed down
with some force, and I

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rotated it through
an angle theta.

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So that's theta and this
is also theta.

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So my question to you, and
we'll have to take out a

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little bit of our trigonometry
skills, is how much did this

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object move up?

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So essentially, what
was this distance?

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What's its distance in the
vertical direction?

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How much did it go up?

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And also, for what distance did
I have to apply the force

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downwards here-- so that's this
distance-- in order for

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this weight to move up this
distance over here?

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So let's figure out
either one.

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So this distance is what?

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Well, we have theta.

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This is the opposite.

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This is a 90 degree
angle, because we

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started off at level.

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So this is opposite.

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And this is what?

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This is the adjacent angle.

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So what do we have there?

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Opposite over adjacent.

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Soh Cah Toa.

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Opposite over adjacent.

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Opposite over adjacent.

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That's Toa, or tangent.

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So in this situation, we know
that the tangent of theta is

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equal to-- let's call
this the distance

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that we move the weight.

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soon.

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So that equals opposite over
adjacent, the distance that we

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moved the weight over 1.

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And then if we go on
to this side, we

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can do the same thing.

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Tangent is opposite
over adjacent.

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So let's call this the distance
of the force.

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So here the opposite of the
distance of the force and the

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adjacent is this 2 meters.

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Because this is the hypotenuse
right here.

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So we also have the tangent of
theta-- now you're using this

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triangle-- is equal to
the opposite side.

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The distance of the force
over 2 meters.

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So this is interesting.

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They're both equal to
tangent of theta.

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We don't even have to
figure out what the

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tangent of theta is.

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We know that this quantity is
equal to this quantity.

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And we can write it here.

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We could write the distance of
the force, that's the distance

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that we had to push down on
the side of the lever

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downwards, over 2, is equal to
the distance of the weight.

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The distance the weight traveled
upwards is equal to

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the distance, the weight,
divided by 1.

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Or we could say-- this
1 we can ignore.

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Something divided
by 1 is just 1.

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Or we could say that the
distance of the force is equal

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to 2 times the distance
of the weight.

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And this is interesting, because
now we can apply what

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we just learned here to figure
out what the force was.

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And how do I do that?

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Well, when I'm applying a
force here, over some

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distance, I'm putting energy
into the system.

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I'm doing work.

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Work is just a transfer of
energy into this machine.

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And when I do that, that
machine is actually

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transferring that energy
to this block.

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It's actually doing work on the
block by lifting it up.

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So we know the law of
conservation of energy, and

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we're assuming that this is a
frictionless system, and that

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nothing is being lost to
heat or whatever else.

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So the work in has to be
equal to the work out.

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And so what's the work in?

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Well, it's the force that I'm
applying downward times the

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distance of the force.

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So this is the work in.

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Force times the distance
of the force.

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I'm going to switch colors
just to keep things

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interesting.

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And that has to be the same
thing as the work out.

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Well, what's the work out?

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It's the force of the weight
pulling downwards.

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So we have to-- it's essentially
the lifting force

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of the lever.

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It has to counteract the force
of the weight pulling

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downwards actually.

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Sorry I mis-said it
a little bit.

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But this lever is essentially
going to be

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pushing up on this weight.

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The weight ends up here.

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So it pushes up with the
force equal to the

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weight of the object.

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So that's the weight of the
object, which is -- I said

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it's a 10 newton object -- So
it's equal to 10 newtons.

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That's the force.

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The upward force here.

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And it does that for
a distance of what?

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We figured out this object, this
weight, moves up with a

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distance d sub w.

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And we know what the distance
of the force is in terms of

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the distance of w.

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So we could rewrite this as
force times, substitute here,

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2 d w is equal to 10 d w.

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Divide both sides by 2 you d w
and you get force is equal to

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10 d w 2 two d w, which is
equaled to, d w's cancel out,

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and you're just left with 5.

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So this is interesting.

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And I think you'll see where
this is going, and we did it

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little complicated this time.

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But hopefully you'll realize
a general theme.

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This was a 10 newton weight.

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And I only had to press down
with 5 newtons in order to

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lift it up.

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But at the same time, I pressed
down with 5 newtons,

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but I had to push down
for twice as long.

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So my force was half as much,
but my distance that I had to

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push was twice as much.

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And here the force is twice as
much but the distance it

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traveled is half as much.

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So what essentially just
happened here is, I

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multiplied my force.

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And because I multiplied
my force, I

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essentially lost some distance.

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But I multiplied my
force, because I

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inputted a 5 newton force.

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And I got a 10 newton force out,
although the 10 newton

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force traveled for
less distance.

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Because the work was constant.

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And this is called mechanical
advantage.

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If I have an input force of 5,
and I get an output force of

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10, the mechanical
advantage is 2.

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So mechanical advantage is equal
to output force over

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input force, and that should
hopefully make a little

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intuitive sense to you.

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And another thing that maybe
you're starting to realize

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now, is that proportion of the
mechanical advantage was

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actually the ratio of this
length to this length.

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And we figured that out by
taking the tangent and doing

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these ratios.

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But in general, it makes sense,
because this force

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times this distance has
to be equal to this

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force times this distance.

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And we know that the distance
this goes up is proportional

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to the length of from the
fulcrum to the weight.

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And we know on this side the
distance that you're pushing

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down, is proportional to the
length from where you're

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applying the weight
to the fulcrum.

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And now I'll introduce you
to a concept of moments.

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In just a moment.

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So in general, if I have, and
this is really all you have to

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learn, that last thought
exercise was just

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to show it to you.

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If I have a fulcrum here, and
if we call this distance d 1

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and we called this
distance d 2.

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And if I want to apply
an upward force here,

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let's call this f 1.

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And I have a downward force,
f 2, in this machine.

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f 2 times d 2 is equal
to d 1 times f 1.

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And this is really all
you need to know.

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And this just all falls
out of the work in is

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equal to the work out.

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Now, this quantity isn't
exactly the work in.

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The work in was this force--
sorry, F2-- is this force

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times this distance.

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But this distance is
proportional to this distance,

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and that's what you
need to realize.

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And this quantity right here is
actually called the moment.

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In the next video, which I'll
start very soon because this

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video is about to end.

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I'm running out of time.

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I will use these quantities to
solve a bunch of mechanical

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advantage problems. See