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# Mechanical advantage (part 3)

## Video transcript

Welcome back. Now let's do some more
mechanical advantage problems. And in this video, we'll focus
on pulleys, which is another form of a simple machine. And we've done some pulley
problems in the past, but now we'll actually understand what
the mechanical advantage inherent in these
machines are. So let me start with a
very simple pulley. So this is the ceiling
up here. I don't know what they call
that part of the pulley. I should learn my actual
terminology. But let's say I have that little
disk where the rope goes over and it rolls so that
the rope can go over it and move without having
a lot of friction. And let's say I have a rope
going over that pulley. That's my rope. And at this end, let's say I
have a weight, a 10-Newton weight, and I'm going to pull
down on this end to make the weight to go up. So my question to you is what is
the mechanical advantage of this system? What is the force that I have to
pull down in order to lift this weight, this 10-Newton
weight in order to produce 10 Newtons of force upwards? Well, in any pulley situation--
and I don't know if textbooks cover it this way,
but this is how I think about it, because you don't
have to memorize formulas. I just think about, well,
what happens to the lengths of rope? Or what is the total distance
that the object you're trying to move travels? And if you know the distance
that it travels versus the distance that you have
to pull, you know the mechanical advantage. So in this situation, if I were
to hold the rope at that point, and if I were to pull
it down 10 feet or some arbitrary distance, what
happens over here? Well, this weight is
going to move up exactly the same amount. Whatever I pull, if I pull a
foot down here, this weight will move up by a foot, so the
distance that I pull here is equivalent to the distance
that it pulls up here. And since we know that the work
in has to equal the work out, we know that the force I'm
pulling down has to be the same as the force or the tension
that the rope is pulling up here. And we could have done that when
we learned about tension, that the tension in the
rope is constant. I'm producing tension in the
rope when I pull here and that's the same pulling force of
the tension on the weight. So this isn't too interesting
of a machine. All it's doing is I pull down
with a force of 10 Newtons and it will pull up with a force
of 10 Newtons, and so the mechanical advantage is 1, no
real mechanical advantage, although this could be useful. Maybe it's easier for
me to pull down than for me to pull up. Or at some point, maybe I can't
reach up here, so it's nice for me to pull down here
where I can reach and the object will keep going up
like in a flag pole or something like that. So this could still be useful
even though its mechanical advantage is only 1. So let's see if we can construct
a pulley situation where the mechanical advantage
is more than 1. So let's say over here at the
top, I still have the same pulley that's attached to the
ceiling, but I'm going to add slight variation here. I have another pulley here. And now let me do the other
pulley down here. And then let me see if I can
draw my rope in a good way. So my rope starts up going up
like that, then it comes back down, comes around the second
pulley, and now this is attached to the ceiling
up here. The second pulley is
actually where the weight is attached to. And let's just call it a
10-Newton weight again, although it doesn't really
matter what the weight is. Let's figure out what the
mechanical advantage is. So the same question. And this is really the question
you always have to ask yourself. If I were take a point on this
rope and if I were to pull it 2 feet down, so let's see I take
this point and I move it 2 feet down, what essentially
happens to the rope? Well, every point on the
rope's going to move 2 feet to the right. I guess you can view it this way
if you view that motion is to the right. But if this length of rope is
getting 2 feet shorter, what is this length of
rope getting? Well, this entire length of rope
is also going to get 2 feet shorter, this entire length
of rope right here. But this entire length of rope
is split between this side-- let me do it in different
color-- between this side and this side. So if I make this side of the
rope shorter-- I mean, the rope goes through the whole
thing, but if I take this side of the rope and I pull
down by 2 feet, what is going to happen? Well, this is going to
get 1 foot shorter. This rope is going to
get 1 foot shorter. This is going to go 1 foot
shorter and this length of the rope is going to get
1 foot shorter. And how do I know that? Well, this is all
the same rope. And if this is getting 1 foot
shorter, and this is one getting 1 foot shorter, it makes
sense this whole thing is getting 2 feet shorter. But the important thing to
realize, if each of these are getting 1 foot shorter,
then this weight is only moving up 1 foot. So when I pull the rope down 2
feet here, this weight only moves up 1 foot. So what is the work
that I'm doing? Well, the work in is the same
as the work out, and we know what the work out is. The work out is going to
be the force that this contraption or this machine is
pulling upwards with, and that's 10 Newtons, so the
workout is equal to 10 Newtons times the distance
that the force is pulling in, times 1 foot. Oh, why did I do feet? I should do meters. That's not a good thing
for me to do. That should be meters. I shouldn't mix English
and metric system. So 10 Newtons times 1 meter,
so it equals 10 joules. And this has to be the
work that I've put into it, too, right? So the work in also has
to be 10 joules. Well, I know the distance
that I pulled down. I know I pulled down 2 meters. So I pulled down 2 meters, so
this has to equal the force times the distance. So the force, which I don't
know, times the distance, which is 2 meters, is
equal to 10 joules. So divide both sides by 2, so
the force that I pulled down with is 5 Newtons. So I pulled down 5 Newtons for
2 meters, and it pulls up a 10-Newton weight for 1 meter. Force times distance is equal
to force times distance. So what was the input force? The input force is equal to 5
Newtons and the output force of this machine is equal
to 10 Newtons. Mechanical advantage is the
output over the input, so the mechanical advantage is equal
to the force output by the force input, which equals
10/5, which equals 2. And that makes sense, because
I have to pull twice as much for this thing to move up
half of that distance. Let's see if we can do another
mechanical advantage problem. Actually, let's do a really
simple one that we've really been working with a long time. Let's say that I have a wedge. A wedge is actually considered
a machine, which it took me a little while to get my
mind around that, but a wedge is a machine. And why is a wedge a machine? Because it gives you mechanical
advantage. So if I have this wedge here. And this is a 30-degree angle,
if this distance up here, let's call this distance
D, what is this distance going to be? Well, it's going to
be D sine of 30. And we know that the sine of 30
degrees, hopefully by this point, is 1/2, so this
is going to be 1/2D. You might want to review the
trigonometry a little bit if that doesn't completely
ring a bell for you. So if I take an object, if I
take a box-- and let's assume it has no friction. We're not going to go into
the whole normal force and all that. If I take a box, and I push it
with some force all the way up here, what is the mechanical
advantage of this system? Well, when the box is up
here, we know what its potential energy is. Its potential energy is going
to be the weight of the box. So let's say this is
a 10-Newton box. The potential energy at this
point is going to be 10 Newtons times its height. So potential energy at this
point has to equal 10 Newtons times the height, which is
going to be 5 joules. And that's also the amount of
work one has to put into the system in order to get it into
this state, in order to get it this high in the air. So we know that we would have
to put 5 joules of work in order to get the box
up to this point. So what is the force that
we had to apply? Well, it's that force, that
input force, times this distance has to equal
5 joules. So this input force-- oh, sorry,
this is going to be-- sorry, this isn't 5 joules. It's 10 times 1/2 times
the distance. It's 5D joules. This isn't some kind of units. It's 10 Newtons times the
distance that we're up, and that's 1/2D, so it's
5D joules. Sorry for confusing you. And so the force I'm pushing
here times this distance has to also equal to 5D joules. I just remembered, I
just used D as a variable the whole time. Dividing both sides by
D, what do I get? The input force had to be
equal to 5 Newtons. I'm dividing both sides
by D meters. So I inputted 5 Newtons of force
and I was able to lift essentially a 10-Newton
object. So what is the mechanical
advantage? Well, it's the force output,
10 Newtons, divided by the force input, 5 Newtons. The mechanical advantage is 2.