Hey, guys. Let's check out this problem here. We've had these 2 blocks that are connected by a cord that goes over a pulley. This is sometimes called an Atwood machine. So we know the masses of the 2 blocks. We know the larger one is 6 and the smaller one is 4. We want to figure out the acceleration of this system and the free body diagram for both objects. It doesn't matter which one I start off with. So if I call this block a and this one block b, then the free body diagrams for this are going to look like the weight. So we have a weight force like this. Then the only other force that is acting on this is going to be the tension from the cord. Right? There's no applied forces and there's no normal or friction or anything like that. Let's move on to the other one, and it's going to look pretty similar. We have a weight force. I'm going to call this b, which is mbg. And then the only other force that is acting on this one is also the tension force. And remember that these are actually action-reaction pairs. This cord that connects them with a pulls on b, b also pulls on a and the tensions actually both act upwards in this case. Alright? So that's our free body diagram, and the next thing we have to do is determine a direction of positive. And so here's the rule. Imagine that these two blocks are kind of just hanging there. We know that the right one, the 6, is heavier. So we know that when you release this thing, one of them is going to have to go up and the other one's going to have to go down. So which one is it? We can kind of just guess or predict that the heavier one has more weight pulling on it. So this is the one that's going to go downwards. So the 6 is going to go down. The 4 is going to go up, which means that the direction of positive is usually going to be the direction that the heavier object is going to fall. So what that means is that our 6-kilogram object is going to go down like this. That's we're going to choose our direction of positive. But because this pulley goes up and over, then if the 6 goes down, the 4 is going to go up. So we're going to choose the upward direction to be positive for the 4-kilogram objects. So our direction of positive is really just the clockwise direction here. That's really important. So now we're going to write F = ma, and we're going to start with the simplest object. But what's lucky for us is that we have both objects that are relatively simple. We only have 2 objects or 2 forces. So I'm going to start off with object a here, and I want to calculate, I want to use my F = mass times acceleration. These are all the forces in the y-axis. Now remember what we said was that for a, anything that's upwards is going to be positive, which means when we expand out our forces, we have the tension that goes in our direction of positive for block 4, for block a. So that means that's going to be positive. And then our MAG points downwards against our positive direction, so it's going to be minus. And this is going to be ma times a. So now we could replace the values that we know. We know tension, so we have tension minus. This is going to be 4 times 9.8, and this is going to equal 4 times a. So I could just simplify real quick. I've got 4, t - 39.2, whoops, equals 4a. So I can't go any further because I want to figure out the acceleration. But remember, I don't know what the tension is. And it's okay. If I get stuck, I'm just going to go to the other objects. So this is going to be block b. So I want to write F = ma for block b now. So remember, now the rule for block b is that the downward direction is going to be positive. So that means that for block b, anything that points downwards is going to be our direction of positive. So for instance, our mbg is actually going to be the positive one here, and our tension points against our direction of positive, so it's going to be minus. So it's really important that you follow those rules. So now we just replace all the values that we know. We know this is 6 times 9.8 minus tension equals 6 times a. So this ends up being 58.8 minus t equals 6a. So, again, I can't go any further because I have this acceleration, but I still need the tension force. So, predictably, I've gotten 2 equations with 2 unknowns. It's usually what happens in these problems. So I'm going to box them, and I'm going to call this one equation number 1 and this one equation number 2. And that brings us to the 4th step. We want to solve for this acceleration. We just have to use either equation addition or substitution, and I'm just going to pick addition. So, basically, what we've got here is we've got 1, which is t - 39.2 equals 4a. Now what I want to do is I want to line up the tension, right, from equation number 2 so that we can add them straight down. So this is going to be 58.8. Looks a little funky, but that's because I'm trying to line up these variables. So I've got these two equations right here. And now remember, you're just going to add them straight down. You just add them straight down like this. And what happens is you're going to cancel out these tensions when you add them. So you add them straight down you get 58.8 and remember this is minus 39.2, so this is minus 39.2 equals and then the 4 +6 is going to be 10a. So this turns out to be 19.6 equals 10a, and that means your acceleration is 1.96 meters per second squared. So let's talk about the direction. We got a positive number for our acceleration. That just means that our acceleration points in our direction of positive. So that means that a is going to be 1.96 downwards, just exactly like we guessed. Alright? So that's it for part a. Now we want to move on to part b, and that's just calculating the tension. And we know if we want to calculate other variables, we're just going to have to plug our a back into our equations, and then to solve for other targets. Right? So we want to figure out we're just going to use one of these equations to solve for t. They both have the same number of terms, but the thing is that this t is actually positive in this equation and this one's negative. So this one is slightly more simple. So that means I'm going to use t - 39.2 equals 4. And then now we know a. Right? So t - 39.2, actually, I'm just going to move this to the other side. So t equals 4 times 1.96, that's what we figured out here, plus 39.2. If you go ahead and work this out, you're going to get 47.04 newtons. That's the answer. That's it for this one, guys. Let me know if you have any questions.
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6. Intro to Forces (Dynamics)
Forces in Connected Systems of Objects
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