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- Welcome back.
- I'm just continuing this ultra, five star problem just
- so you make sure you can understand everything that's
- going on, especially with tension and pulleys and all
- the rest.
- So here we were.
- We figured out the net forces acting on this block.
- And we said it was the tension of the rope minus
- the force of friction.
- And we figured out the force of friction in
- the traditional way.
- And that equals the mass of the block times the
- acceleration of the block.
- And actually, it would have been a good idea here to solve
- for tension.
- So what is the tension pulling on this block?
- The tension is equal to 10 times the acceleration of the
- block plus 39.2 newtons.
- And actually, this is probably the one that I should have
- squared because this is even more useful than
- what I wrote before.
- So we figured out the tension of the rope here in terms of
- the acceleration of the block.
- And then we moved on to the pulley.
- And we said, OK, well what are the net forces acting on here?
- We have 150 newtons to the right, and we have twice the
- tension pulling back right?
- Cause we have the tension here pulling back
- and the tension there.
- That's kind of what pulleys do.
- And we knew that because the tension
- in the rope is constant.
- And if you were to do it here, not that it matters, the
- tension of the rope here is T and then the wall is pulling
- back on the rope with the force T.
- And that's why this point is constant.
- But we don't worry about that, so back to the problem.
- So we said the net forces on the pulley, which is 150 minus
- 2 times the tension, is equal to the mass of the pulley, 5
- kilograms, times the acceleration-- and this is the
- p here-- times the acceleration of the pulley.
- And at the very beginning of the problem I had said, well
- the acceleration of the pulley is half the
- acceleration of the block.
- And I went off about all the reasons.
- And I'll say it one more time, because if this pulley moves
- an inch to the right, not only will the pulley move an inch
- to the right, this length of wire will get an inch longer,
- right, cause this is stationary here.
- So this length of wire, since the wire is a constant length,
- will get an inch shorter.
- So this is getting an inch shorter and the pulley is
- moving an inch to the right, so this block will move 2
- inches to the right.
- So that's how we came to the conclusion that whatever the
- velocity to the right of this pulley is, the velocity of
- this block will be twice that.
- Or whatever the acceleration of this pulley is to the
- right, the acceleration of this block will be twice that.
- And that's what we wrote here.
- The acceleration of the pulley or the acceleration of the
- block is twice the acceleration of the pulley.
- Or the acceleration of the pulley is 1/2 the acceleration
- of the block.
- So with that said, we took this equation and we
- substituted the acceleration of the pulley.
- And we said well the acceleration of the pulley is
- just half the acceleration of the block.
- The pulley's velocity and its acceleration is half whatever
- the block's is.
- And so we substituted that in for the
- acceleration of the pulley.
- And now we can substitute tension, which we solved here,
- for the-- my mind's getting flustered-- for the tension in
- this equation.
- So what do we get?
- We get 150-- let me move back to a nice color-- minus 2
- times this expression.
- This is the tension in the rope.
- 10 times a, b, acceleration of the block, plus 39.2 is equal
- to 5 times 1/2.
- Well what's 5 times 1/2?
- It's 2.5 acceleration of the block.
- So we have 150 minus 20 times the acceleration of the block
- minus-- what is this-- 78.4 is equal to 2.5 times the
- acceleration of the block.
- Well let's add 20 times the acceleration of the block to
- both sides.
- Well, we'll just simplify this.
- 150 minus 78.4 is equal to 71.6.
- So 150 minus 78.4 is 71.6.
- So that's just this minus this.
- And I'm going to add 20 a b to both sides, so I'm essentially
- moving this 20 a b over on that side.
- I'm skipping a couple of steps just to save space.
- So if I add 20 times the acceleration of the block.
- That's not a, b.
- It's a sub b.
- You get 22.5 times the acceleration of the block.
- And this is a 6, not a block.
- 71.6 is equal to 22.5 times the acceleration of the block.
- We're almost done.
- So we divide both sides by 22.5 and what do we get?
- Let's see.
- Divided by 22.5, and we get the acceleration of the block.
- So that's this block right here.
- The acceleration of the block is 3.18
- meters per second squared.
- That's how fast this thing accelerates to the right.
- And we already figured out that the pulley accelerates at
- half that rate.
- So whatever that number divided by 2.
- So the pulley itself, the acceleration of the pulley is
- half of this number, which is 1.59 meters per second squared
- to the right.
- I know this was a fairly difficult problem and the key
- realizations though, I think that you had to discover or
- realize in order to be able to do this
- was a couple of things.
- One, that the acceleration of this pulley is half the
- acceleration of the block, or the acceleration of the block
- is double the acceleration of the pulley.
- You have to realize that.
- And then you just have to work out the net horizontal forces
- and realize well the only thing pulling on this block is
- friction going backwards and tension of
- the rope to the right.
- And that same tension of the rope is constant throughout
- this wire or through this rope.
- And on this pulley, and this might not be something that
- you had already realized about pulleys, but now you will, is
- that since the rope essentially goes around the
- pulley, it's pulling twice on the pulley.
- And the tension's constant, so it's pulling with a tension of
- T on the top and the bottom.
- And if you realized those things, then it's just a
- little bit of algebra to get the acceleration of the block
- and the acceleration of the pulley.
- Anyway, I hope I didn't confuse you too much.
- I would call this a five star problem if you would see this
- type of problem on physics competitions.
- So if you know how to do this, you're doing well, at least as
- far as tension and ropes and friction are concerned.
- I'll see you in the next video.

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