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# Moving pulley problem (part 1): What happens when we pull on a pulley and the pulley is pulling on other things?

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- Welcome back.
- We'll now do a problem that I think will bring everything
- together and frankly, this is a fairly difficult problem.
- And if you get this problem, you are on your way to
- becoming a physics master.
- So let's get started with this problem.
- So we have this set up here, where we have
- a wall and a floor.
- And we have a wire or rope attached to the wall here.
- And it goes around this pulley.
- And this pulley, for the sake of this problem, let's just
- say this pulley is somehow supported in the vertical
- direction, so it doesn't fall to the ground.
- And then this wire goes back around the pulley and then
- it's attached to this 10 kilogram mass.
- And in this case-- and this is the first time we're dealing
- with a pulley that actually has mass-- this pulley has a
- mass of 5 kilograms.
- And even more, the pulley isn't even fixed.
- We are going to pull on the pulley.
- And we were to pull on the pulley with a force-- let me
- get my pen tool set up because this is an exciting problem.
- We will pull on the pulley with a force of-- let me make
- it clean lines because-- let's say we will pull on this
- pulley with a force of 150 Newtons.
- Somehow it's being compelled to go in that direction with a
- force of 150 Newtons.
- Even more, the coefficient--
- So this pulley is somehow up in the air, and it's being
- supported there.
- So we're not going to assume that it falls or
- something like that.
- It's only being pulled in the rightward direction, so don't
- worry about the up and down motion of it.
- But the problem also tells us that the coefficient of
- friction between this block and the ground, the
- coefficient of friction is 0.4.
- So I could write that here.
- So the question is, what happens here?
- What happens?
- So in any physics problem, before you break into the
- numbers, I think it's always a good idea, just to get a
- conceptual sense of what it should look like, what you
- would think would happen.
- So someone's pulling on this pulley and this
- wire's staying fixed.
- So what's going to happen?
- They're going to pull on this pulley.
- The pulley's going to go more and more to the right.
- As the pulley goes more and more to the right, this top
- length of wire will get longer and the only way that that top
- length of wire gets longer is if this bottom length of wire
- gets shorter.
- So the pulley will be moved to the right and at the same
- time, this 10 kilogram mass will get closer and closer to
- the pulley.
- And if you think about it, whatever the velocity of the
- pulley to the right, the velocity of this 10 kilogram
- mass would be twice that.
- Why is that?
- Well let's say this pulley moves 1 inch
- to the right, right?
- If the pulley moves 1 inch to the right, then this length of
- rope right here will get 1 inch longer, right?
- Because the pulley went 1 inch to the right.
- And this length of rope-- because the rope is of a
- constant length, we're assuming it doesn't stretch or
- anything-- will get 1 inch shorter.
- So not only did the pulley move 1 inch to the right, but
- this rope down here got 1 inch shorter.
- So this block would have moved 2 inches to the right.
- Hopefully that makes sense.
- So whatever the velocity is of this pulley, this mass's
- velocity will be twice as much.
- So similarly, whatever the acceleration of this pulley to
- the right, the acceleration of this block
- will be twice as much.
- And why is that?
- Well acceleration is just change in velocity and so, if
- my velocity has doubled and this guy's velocity is double
- that, his change in velocity will be double that.
- I don't know if I just said a circular statement.
- But hopefully that makes sense.
- Just think about what happens.
- If this guy moves an inch, this length of cord will also
- get shorter by an inch.
- So the pulley will move an inch, and then this guy will
- get closer to a pulley, so he will have moved 2 inches.
- So his velocity and acceleration are double that
- of the pulley.
- And that's an important thing to realize.
- So with that out of the way, let's solve the problem.
- That really is kind of the big thing you should realize about
- this problem.
- So with that tucked away in the back of our mind, let's do
- the problem.
- So what are all the forces acting on
- this block right here?
- Well, we know it's going to be moving towards the right.
- It's actually going to be moving towards the right twice
- as fast as the pulley.
- So if it's moving to the right, which direction is the
- force of friction acting in?
- Well the force of friction is always the spoiler.
- It's always going in the opposite direction.
- So you have the force of the friction going backwards.
- Right?
- And what is that force of friction?
- Well it's going to be the weight of this mass, this
- weight of this block, times the coefficient of friction.
- Because the weight comes down and that's equal to the normal
- force, and the normal force is equal to weight.
- You know all of this already.
- So the force of friction is going to be equal to the
- coefficient of friction, 0.4, times the
- weight of this block.
- And what's the weight of this block?
- It's going to be 10 kilograms, which is the mass, times the
- acceleration of gravity.
- So that's 9.8.
- So it's 98 Newtons.
- I'm skipping some steps here because I think things like,
- what's the weight of a 10 kilogram mass block, I think
- are, hopefully, a bit of second nature to you now.
- So it's 0.4 times 0.98 Newtons.
- And if we get the calculator out, that is
- 0.4 times 0.-- whoops.
- It's 0.4 times 98 equals 39.2 Newtons.
- So that's the force of friction going
- backwards on this mass.
- Well what's compelling the block to go to the right?
- Well, it's this wire, this rope, right?
- So the tension in this rope is pulling on this block, so we
- call that T.
- All right?
- So what are the net forces acting on this block?
- And let's say, what are the net forces to the right?
- Well it's T, right?
- Which is the force of tension of this rope.
- I could've done it along the rope, but I just did it here.
- T is a force of tension.
- I'll draw it-- actually, I will draw it along the rope.
- Just so you know, it's the rope that's exerting this
- force of tension.
- So the force of tension minus the force of friction, which
- we figured out to be 39.2 Newtons, is equal to the net
- force on this object, right?
- Is equal to the net force, At.
- Least in the left, right or horizontal
- direction on this object.
- And that's going to be equal to its mass times its
- acceleration, right?
- The net forces on an object are equal to its mass times
- acceleration.
- That's Newton's second Law.
- So what's its mass?
- It's 10.
- And if we knew the acceleration, we wouldn't have
- to do this problem.
- So we don't know what it is, so let's just say it's a.
- So 10 times acceleration is a.
- So these are the net forces acting on this mass.
- And we don't know what a is.
- So let's put that aside a little bit.
- Maybe I'll put a little square around it.
- Let's figure out what's happening
- to the pulley itself.
- So this is interesting.
- The pulley has this mysterious force, maybe it's my hand,
- pulling at 150 Newtons to the right.
- And what's pulling it to the left?
- Well, in both cases, this wire is pulling it to the left.
- And the tension throughout a wire is constant, unless the
- material changes or something-- well even then it
- shouldn't change.
- So the tension through the wire is constant.
- And in both cases, the wire is pulling back on this pulley.
- So if the tension here is T, going in this direction, the
- tension in this direction is T.
- And the tension in this direction is also T.
- And we'll learn more about pulleys when we start doing
- mechanical advantage and things, and how it doubles the
- force needed, but the distance goes in half, and all of these
- type of things.
- And we'll do that later.
- But all you have to realize is that the tension through the
- wire is constant.
- And this pulley, essentially, has the wire pulling on it
- twice, right?
- Once on top and once on the bottom.
- So what are the net forces acting on this pulley?
- Well you have 150 Newtons to the right.
- And then it has the force of tension twice pulling to the
- left, right?
- Because the wire wraps around it.
- So minus 2T.
- And that equals the mass times the
- acceleration of this pulley.
- So that's the mass, which is 5 kilograms, times the
- acceleration of pulley.
- I'll write the acceleration of pulley.
- And this one up here was the acceleration of the block.
- What was the first thing that we
- discovered about this problem?
- That the acceleration of the pulley is equal to half the
- acceleration of the block.
- Or, that the acceleration of the block is two times the
- acceleration of the pulley.
- Either way.
- That was the first thing that I went off about, how the
- pulley moves an inch, and at the same time, this bottom
- part of the rope or wire gets shorter by an inch.
- So this thing moves twice as fast. So let's substitute here
- for the acceleration of the block.
- So the acceleration of the pulley is 1/2 times the
- acceleration of the block.
- So let's substitute.
- So we get 150 Newtons minus 2 times the tension of the wire
- is equal to 5 times-- instead of the acceleration of the
- pulley, it's 1/2 the acceleration of the block, of
- this block here.
- And now let's try to solve.
- Well, we don't want to solve yet.
- We have to somehow substitute for what the
- tension of the wire is.
- So let's do that.
- And actually, I'm getting flustered because I realize I
- only have 10 seconds left.
- So I will see you in the next video.

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