Mechanical advantage (part 3)

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.