Mechanical Advantage (part 3) Introduction to pulleys and wedges
Mechanical Advantage (part 3)
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- 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.
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