Welcome back.
We'll now use a little bit of what we've learned about work
and energy and the conservation of energy and
apply it to simple machines.
And we'll learn a little bit about mechanical advantage.
So I've drawn a simple lever here.
And you've probably been exposed to
simple levers before.
They're really just kind of like a seesaw.
This place where the lever pivots.
This is called a fulcrum.
Just really the pivot point.
And you can kind of view this as either a seesaw or a big
plank of wood on top of a triangle, which essentially is
what I've drawn.
So in this example, I have the big plank of wood.
At one end I have this 10 newton weight, and I've
written 10 in there.
And what we're going to figure out is one, how much force--
well, we could figure out a couple of things.
How much force do I have to apply here to
just keep this level?
Because this weight's going to be pushing downwards.
So it would naturally want this whole
lever to rotate clockwise.
So what I want to figure out is, how much force do I have
to apply to either keep the lever level or to actually
rotate this lever counterclockwise?
And when I rotate the lever
counterclockwise, what's happening?
I'm pushing down on this left-hand side, and I'm
lifting this 10 newton block.
So let's do a little thought experiment and see what
happens after I rotate this lever a little bit.
So let's say, what I've drawn here in mauve, that's our
starting position.
And in yellow, I'm going to draw the finishing position.
So the finishing position is going to look
something like this.
I'll try my best to draw it.
The finishing position is something like this.
And also, one thing I want to figure out, that I wanted to
write, is let's say that the distance, that this distance
right here, from where I'm applying the force to the
fulcrum, let's say that that distance is 2.
And from the fulcrum to the weight that I'm lifting, that
distance is 1.
Let's just say that, just for the sake of argument.
Let's say it's 2 meters and 1 meter, although it could be 2
kilometers and 1 kilometer, we'll soon see.
And what I did is I pressed down with some force, and I
rotated it through an angle theta.
So that's theta and this is also theta.
So my question to you, and we'll have to take out a
little bit of our trigonometry skills, is how much did this
object move up?
So essentially, what was this distance?
What's its distance in the vertical direction?
How much did it go up?
And also, for what distance did I have to apply the force
downwards here-- so that's this distance-- in order for
this weight to move up this distance over here?
So let's figure out either one.
So this distance is what?
Well, we have theta.
This is the opposite.
This is a 90 degree angle, because we
started off at level.
So this is opposite.
And this is what?
This is the adjacent angle.
So what do we have there?
Opposite over adjacent.
Soh Cah Toa.
Opposite over adjacent.
That's Toa, or tangent.
So in this situation, we know that the tangent of theta is
equal to-- let's call this the distance
that we move the weight.
soon.
So that equals opposite over adjacent, the distance that we
moved the weight over 1.
And then if we go on to this side, we
can do the same thing.
Tangent is opposite over adjacent.
So let's call this the distance of the force.
So here the opposite of the distance of the force and the
adjacent is this 2 meters.
Because this is the hypotenuse right here.
So we also have the tangent of theta-- now you're using this
triangle-- is equal to the opposite side.
The distance of the force over 2 meters.
So this is interesting.
They're both equal to tangent of theta.
We don't even have to figure out what the
tangent of theta is.
We know that this quantity is equal to this quantity.
And we can write it here.
We could write the distance of the force, that's the distance
that we had to push down on the side of the lever
downwards, over 2, is equal to the distance of the weight.
The distance the weight traveled upwards is equal to
the distance, the weight, divided by 1.
Or we could say-- this 1 we can ignore.
Something divided by 1 is just 1.
Or we could say that the distance of the force is equal
to 2 times the distance of the weight.
And this is interesting, because now we can apply what
we just learned here to figure out what the force was.
And how do I do that?
Well, when I'm applying a force here, over some
distance, I'm putting energy into the system.
I'm doing work.
Work is just a transfer of energy into this machine.
And when I do that, that machine is actually
transferring that energy to this block.
It's actually doing work on the block by lifting it up.
So we know the law of conservation of energy, and
we're assuming that this is a frictionless system, and that
nothing is being lost to heat or whatever else.
So the work in has to be equal to the work out.
And so what's the work in?
Well, it's the force that I'm applying downward times the
distance of the force.
So this is the work in.
Force times the distance of the force.
I'm going to switch colors just to keep things
interesting.
And that has to be the same thing as the work out.
Well, what's the work out?
It's the force of the weight pulling downwards.
So we have to-- it's essentially the lifting force
of the lever.
It has to counteract the force of the weight pulling
downwards actually.
Sorry I mis-said it a little bit.
But this lever is essentially going to be
pushing up on this weight.
The weight ends up here.
So it pushes up with the force equal to the
weight of the object.
So that's the weight of the object, which is -- I said
it's a 10 newton object -- So it's equal to 10 newtons.
That's the force.
The upward force here.
And it does that for a distance of what?
We figured out this object, this weight, moves up with a
distance d sub w.
And we know what the distance of the force is in terms of
the distance of w.
So we could rewrite this as force times, substitute here,
2 d w is equal to 10 d w.
Divide both sides by 2 you d w and you get force is equal to
10 d w 2 two d w, which is equaled to, d w's cancel out,
and you're just left with 5.
So this is interesting.
And I think you'll see where this is going, and we did it
little complicated this time.
But hopefully you'll realize a general theme.
This was a 10 newton weight.
And I only had to press down with 5 newtons in order to
lift it up.
But at the same time, I pressed down with 5 newtons,
but I had to push down for twice as long.
So my force was half as much, but my distance that I had to
push was twice as much.
And here the force is twice as much but the distance it
traveled is half as much.
So what essentially just happened here is, I
multiplied my force.
And because I multiplied my force, I
essentially lost some distance.
But I multiplied my force, because I
inputted a 5 newton force.
And I got a 10 newton force out, although the 10 newton
force traveled for less distance.
Because the work was constant.
And this is called mechanical advantage.
If I have an input force of 5, and I get an output force of
10, the mechanical advantage is 2.
So mechanical advantage is equal to output force over
input force, and that should hopefully make a little
intuitive sense to you.
And another thing that maybe you're starting to realize
now, is that proportion of the mechanical advantage was
actually the ratio of this length to this length.
And we figured that out by taking the tangent and doing
these ratios.
But in general, it makes sense, because this force
times this distance has to be equal to this
force times this distance.
And we know that the distance this goes up is proportional
to the length of from the fulcrum to the weight.
And we know on this side the distance that you're pushing
down, is proportional to the length from where you're
applying the weight to the fulcrum.
And now I'll introduce you to a concept of moments.
In just a moment.
So in general, if I have, and this is really all you have to
learn, that last thought exercise was just
to show it to you.
If I have a fulcrum here, and if we call this distance d 1
and we called this distance d 2.
And if I want to apply an upward force here,
let's call this f 1.
And I have a downward force, f 2, in this machine.
f 2 times d 2 is equal to d 1 times f 1.
And this is really all you need to know.
And this just all falls out of the work in is
equal to the work out.
Now, this quantity isn't exactly the work in.
The work in was this force-- sorry, F2-- is this force
times this distance.
But this distance is proportional to this distance,
and that's what you need to realize.
And this quantity right here is actually called the moment.
In the next video, which I'll start very soon because this
video is about to end.
I'm running out of time.
I will use these quantities to solve a bunch of mechanical
advantage problems. See