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Two masses hanging from a pulley

In this video David explains how to find the acceleration of two masses hanging from a pulley (using the easy method). Created by David SantoPietro.

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Video transcript

- [Instructor] Let's solve some more of these systems problems. If you remember, there's a hard way to do this, and an easy way to do this. The hard way is to solve Newton's second law for each box individually, and then combine them, and you get two equations with two unknowns, you try your best to solve the algebra without losing any sins, but let's be honest, it usually goes wrong. So, the easy way to do this, the way to get the magnitude of the acceleration of the objects in your system, that is to say, if I wanna know the magnitude at which this five kilogram box accelerates, or that this three kilogram box accelerates, all I need to do is take the net external force that tries to make my system go, and then I divide by my total mass of my system. This is a quick way to get what the magnitude of the acceleration is of the objects in my system, but it's good to note, it'll only work if the objects in your system are required to move with the same magnitude of acceleration. And in this case they are, what I have here is a five kilogram mass tied to a rope, and that rope passes over a pulley, pulls over and connects to this three kilogram mass so that if this five kilogram mass has some acceleration downward, this three kilogram mass has to be accelerating upward at the same rate, otherwise this rope would break or snap or stretch, and we're assuming that that doesn't happen. So this rope is the condition that requires the fact that this rope doesn't break is what allows us to say that the system is just a single, big total mass with external forces exerted on it. So how would we solve this? I'd just say that, well, what are the external forces? Keep in mind, external forces are forces that are exerted on the objects in our system from objects outside of our system. So one external force would just be the force of gravity on this five kilogram mass. So I'm gonna have a force of gravity this way, and that force of gravity is just going to be equal to five kilograms times 9.8 meters per second squared, because that's how we find the force of gravity. Should I make it positive or negative? Well, this five kilogram is gonna be the one that's pulling downward, so if the question is, I hold these masses and I let go, what's the acceleration? This five kilogram mass is gonna accelerate downward, it's gonna drive the system forward. That's the force making the system go, so I'm gonna make that a positive force. And then I figure out, are there any other forces making this system go? No, there are not. You might say, well what about this tension over here? Isn't the tension on this three kilogram mass? Isn't that tension making this system go? Not really, because that's an internal force exerted between the objects in our system and internal forces are always opposed by another internal force. This tension will be pulling the three kilogram, trying to make it move, but it opposes the motion of the five kilogram mass, and if we think of this three plus five kilogram mass as a single object, these end up just canceling on our single object that we're viewing as one big eight kilogram mass. So those are internal forces. We don't include them, they're not part of this trick. We have to figure out what other forces would try to make this system go or try to prevent it from moving. Another force that tries to prevent it from moving is the force of gravity on the three kilogram mass. Or, one force that tries to prevent the system from moving would be this force of gravity. How big is that? That's three kilograms times 9.8 meters per second squared. And that's trying to prevent the system from moving. This five kilogram mass is accelerating downward, and this force is in the opposite direction of motion. That trips people out sometimes. They're like, I don't understand, they're both pointing down. Shouldn't they have the same sign? They would when we're using Newton's second law the way we usually use it, but when we're using this trick, what we're concerned with are forces in the direction of motion, this is an easy way to figure it out, forces in the direction of motion we're gonna call positive. And any forces opposite the direction of motion we're gonna call negative. So, forces that propel the system forward we'll just call that positive direction. Forces that resist the motion, we're just gonna call that the negative direction. And since this is on this side of the motion of the system, this system is, everything in this system is going this way. The three kilogram mass goes up. The string over here goes up. The string up here goes to the right. The string right here goes down. The five kilogram mass goes down. Because all the motion in the system is this way, we'd find that way's positive, but this force of gravity on the three kilogram mass is the opposite direction. It's opposing the motion of the system. It's preventing the system from accelerating as fast as it would have. That's why we subtract it. And now we just divide by the total mass. And the total mass is just five plus three, is gonna be eight kilograms, and I get the acceleration of my system. So if I just add this up, I get 2.45 meters per second squared. So this is a really fast way to get what the acceleration of our system is, but you have to be careful. If the question is, what's the acceleration of a five kilogram box? Well, technically, that acceleration of the five kilogram box would be negative 2.45. What we really found here, since we were just finding the magnitude, was the size of the acceleration, since this five kilogram box is accelerating down, and we usually treat down as negative. You won't wanna forget that negative in putting in that answer the acceleration of the three kilogram box, however, would be positive 2.45 meters per second squared. So when you're applying this to an individual box, you have to be very careful and make sure that you apply that acceleration with the correct sign for that particular box. And if you wanted to find the tension now, now it's easy to find the tension. I could find this tension right here if I wanted to. If the next step was find the tension in the string connected to the boxes, now I can just use Newton's second law, but the way we always use it. I'm done with the trick. The trick is just the way to get the magnitude of the acceleration. Now that I have that, I'm done treating it as a system or a single object. I'll look at this single five kilogram mass all alone, and I'll say that the acceleration of the five kilogram mass, which is Newton's second law, is gonna equal the net force on the five kilogram mass divided by the mass of the five kilogram mass. I know the acceleration of the five kilogram mass, but if I'm gonna treat up as positive now, I gotta plug this acceleration in with a negative sign. So negative 2.45 meters per second squared is gonna equal the net force on the five kilogram mass. I've got tension up, you might be like, wait, we said that was an internal force. It was an internal force, and we didn't include it up here, but we're doing the old rules now. Normal second law in the vertical direction. So I use vertical forces, and if they're upward I'm gonna treat them as positive, and if they're downward like this five times 9.8, I'm gonna treat it as a negative, because it points down. Five times 9.8 meters per second squared, and I divide by the five kilogram mass, 'cause that's the box I'm analyzing. I'm not analyzing the whole system. I'm just analyzing the five kilogram box now. And I can solve and I can get my tension. The alternate way to do this would be to say, all right, let's just treat down as positive for this five kilogram mass. I'd then plug my acceleration in as positive, and I'd plug my force of gravity in positive, then my tension would be negative. I'd get the same value. Here I'm just solving for the magnitude of the tension anyway. So if I solve this, if I plug this into the calculator and solve for tension, I'm gonna get 36.75 Newtons, which is less than the force of gravity, which it has to be, 'cause if the tension was greater than the force of gravity, this five kilogram mass would accelerate up. We know that doesn't happen. The tension's gotta be less than the force of gravity, so that this five kilogram mass can accelerate downward. So that's a quick way to solve for the magnitude of the acceleration of the system by treating it as a single object. We're saying that if it's a single object, or thought of as a single object, which we can do, 'cause these are required to have the same acceleration, or same magnitude of the acceleration, that if we're treating it like a single object, only external forces matter, and those external forces that make the system go are going to accelerate the system. And those external forces that resist the motion are trying to reduce the acceleration, and we divide by the total mass of the system that we're treating as one object, we get the acceleration. If that still seems like mathematical witchcraft, or if you're not sure about this whole idea, I encourage you to go back and watch the video. We solved one of these types of problems the hard way. And you see, you really do end up with the force that tries to make the system go externally, and the external force that tries to stop it divided by the total mass gives you the acceleration. Essentially, what we're saying is that these internal forces cancel if you're thinking of this system as one single object, 'cause these are applied internally, and they're opposed to each other. One tries to make the system go, one tries to make the system stop.