Introduction to mechanical advantage

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.

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