Atwood Machine

by Patrick Ford
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Hey, guys, let's check out this problem here. We have these two 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 two blocks. One of the larger one is six and smaller ones for We want to figure out the acceleration of this system of connected objects. So we want to figure out what A is equal to. So, luckily for us, we're just gonna go ahead and stick to the steps, right? We can figure out the we're gonna draw the free body diagrams for both objects. 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 is going to look like the weight we have weight force like this, then the only other force that's acting on this is gonna be the tension from the cord, right? There's no plaque 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 gonna call this B, which is MBG and then the only other forces 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. All right, so that's our free body diagram. And the next thing we have to do is determine the 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 six is heavier. So we know that when you release this thing, one is gonna have to go up and the other one is 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 gonna go downwards. So the six is gonna go down, the four is gonna 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 kg object is going to go down like this, that we're going to choose our direction of positive. But because this pulley goes up and over than if the six goes down, the four is going to go up. So we're gonna choose the upward direction. To be positive for the 4 kg objects or direction of positive is really just the clockwise direction here. That's really important. So now we're gonna write f equals M, and we're gonna start with the simplest object. But what's lucky for us is that we have both objects that are relatively simple. We only have two objects or two forces. So I'm going to start off with object A here, and I want to calculate. I want to use my F equals 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 four for Block A. So that means that's going to be positive. And then our MSG points downwards against our positive direction. So it's gonna be minus and this is gonna be m a times a. So now we can replace the values that we know. We know 10. So we have tension minus. This is gonna be four times 9.8 and this is going to equal four times a So I could just simplify real quick. I've got four t minus 39.2. Whoops equals for a 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 gonna go to the other objects. So this is gonna be blocked. Be so I want to write f equals M A 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, are MBG is actually going to be the positive one here, and our tension points against our direction of positive, So it's gonna be minus, so it's really important that you follow those rules. So we just replace all the values that we know. We know this is six times 98 minus tension equals six times a So this ends up being 58.8 Minus T equals six A. 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 two equations with two unknowns. Usually what happens to these problems? So I'm going to box them, and I'm gonna call this one equation number one, and this one equation number two embraces the fourth step. We want to solve this acceleration. We just have to use either equation, addition or substitution. And I'm just gonna pick addition. So basically, what we've got here is we've got one, which is t minus 39.2 equals four A. Now what I want to do is I want to line up the tension right from the equation number two so that we can add them straight down. So this is gonna 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 remember, you're just going to add them straight down. Just add them straight down like this. And what happens is you're gonna cancel out these tensions when you add them. So when you add them straight down, you get 58.8. Remember, this is a minus 39.2, so this is minus 39.2 equals and then four plus six is going to be A. So this turns out to be 19. 6 equals 10 A. And that means your acceleration is 1.96 m per second squared. So let's talk about the direction we got a positive number for 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. All right, 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 gonna have to plug our A back into the 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 amount of terms. But the thing is that this tea is actually positive in this equation and this one is negative. So this one is slightly more simple. So that means I'm gonna use t minus 39.2 equals four. And then now we know a right. So in T minus 39.2. Actually, I'm just going to do this the other side so t equals four times 1.96. That's what we figured out here, plus 39.2. And if you go ahead and work this sound, you're going to get 47.4 Newton's and that's the answer. That's it for this one. Guys, let me know if you have any questions.