7. Friction, Inclines, Systems

Stacked Blocks

# Stacked Blocks

Patrick Ford

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Hey guys. So we've become really familiar with connected systems of objects and also friction. And in this video, I want to talk about a specific type of problem where you have objects that are connected not by ropes or cables, but they're basically just stacked on top of each other. I like to call these problems stacked blocks problems. And this is really interesting things that happened here. So let's go ahead and take a look. Now we're gonna come back to this point in just a second here. We're gonna skip and start with the example, because it's actually very easy to understand what's happening here. The idea is that we have these two blocks. One is 10 and 5 kg. I'll call this one a and this one be. So the idea here is that the floor is frictionless. There's no friction on the bottom surface, But between the two blocs, there is going to be friction. So what happens is I'm gonna pull the bottom block with some force, which means I'm gonna give it some acceleration. Notice how the two blocks stay together. What I want to do is I want to figure out the maximum acceleration that I can give the bottom block so that the two boxes remain moving together. So here's what this means. You pull, there's some acceleration, but the boxes stay together. You pull harder, there's more acceleration. The boxes still stay together. Eventually you can pull hard enough. You can give this an acceleration fast enough that the box is actually starts sliding relative to each other. They are no longer moving together. I want to figure out the maximum acceleration that I can do that with. Right. So basically, this is just gonna be like any other connected objects with system with friction type problem. We're gonna draw the free body diagrams for both. Let's go ahead and get started here. So here I've got A and B the free body diagram for A is gonna look like this. Actually, let me scoot this down a little bit. So I've got my weight force. This is Maggie, and then I have any applied forces or tensions. But there's no applied force that's acting on the top block. Remember, I'm only pulling on the bottom block here, so when I draw this free body diagram, there's no f so I've got a normal force, though, because I've got these two surfaces in contact. One way you can think about this is that a the weight force pushes down on B, so there's a reaction force to that surface push. So I'm gonna call this the normal between the two objects. NBA. Now, finally, what happens is we look at friction, right? So what happens is here you've got this block and it's moving to the rights. So there has to be a force that's acting on the top block to keep it accelerating and moving to the right as well. And that force is the force of friction. Remember that friction tries to stop or prevent the velocity between the two surfaces. So we're pulling on the bottom block, it's moving, and friction actually keeps the top block moving as well. So when you have objects that are stacked on top of each other, the force that acts on the top object that causes it to move is the force of static friction. All right. And so what's interesting about this static friction here? We know this is f s. Is that notice that the velocity of the system is also going to be to the right. So unlike for previous problems, where velocity and friction are always pointing opposite directions, the friction actually acts in the same direction as the direction of motion. All right, the last thing I want to talk about is we have to remember that when we talked about static and kinetic friction, we basically said moving versus not moving We could be a little bit more specific here because now these two surfaces can possibly move relative to each other. So to be more specific, the friction is going to be kinetic. Whatever the relative velocity between the two surfaces is not zero. Anytime you have these two surfaces that are sliding relative to each other, there's gonna be some kinetic friction. Friction is static. On the other hand, whenever the relative velocity between the two surfaces is equal to zero, basically, any time they are moving together like this, that's gonna be static friction. All right, so let's go ahead and get started here. We already know the type of friction that we're dealing with. Um, we actually have to go ahead and draw a free body diagram for B. So let's go and do that real quick. You've got the weight forest that acts on B and now we've got the applied Force. This is my f. We have the normal force that's acting from the floor. This is from the floor on to be so I'm gonna call this n B. But there's also another normal force that's basically part of an action reaction pair be pushes on upwards on a so a pushes downwards on B because of action reaction. So this is a downwards force and I'm going to call this n a B. There's one more action reaction pair, though as well. So I remember we said that there's friction between the two surfaces. There's gonna be a friction force to the right on the top block because of action reaction, there's an equal and opposite force that acts on the bottom blocks on the bottom block. There's another friction force that acts to the left. So really, these two friction forces are actually going to be the same. These are the same fss. Alright, so now we're gonna go ahead and write our F equals m A. Because we're trying to figure out an acceleration so If we're trying to figure out the acceleration, we're gonna start with the simplest object. So basically, we're gonna start off with, uh, the object A So we've got the sum of all forces and really there's only one force to consider, and I'm gonna choose the right direction to be positive. So I got this f s the static friction here, and I'm trying to figure out the maximum acceleration. So if I figure out the maximum acceleration right before the object starts sliding relative to each other, then I'm not just going up against any old friction or static friction. This is actually gonna be f s. Max. What I'm looking for here is the acceleration where the static friction is going to be maximum. So we have an equation for this. So we have, um, UK, or sort of you static times the normal. But what normal are we using in this problem? There's actually three. There's N B A and A B and NB. There's a whole bunch of normals and these problems, which one do we use? Well, basically, the idea is that if you're looking for the friction between the two surfaces Fs Max you're just going to use the normal between the two surfaces. This is going to be the normal force between the two surfaces or between the blocks, right? So really, we're just gonna use em. Use static times N b A, and that's going to equal to mass times a max. So if we can actually go ahead and solve for N B A. Um, if you think about this, these two forces, the weight and the normal have to cancel each other because the top block isn't accelerating vertically. So this NBA here is really just the weight force. So that means we have new static times, M a G equals a times a max. And if you can see here, what happens is that the mass of block A actually cancels out from the equation. And this a max is just gonna equal mu static times gravity. So it's really just going to be 0.7 times 9.8, and the acceleration maximum is going to be 6.86 m per second square. This is the maximum acceleration anything faster than 6 86 and the blocks now starts sliding relative to each other and the friction actually becomes kinetic. So we don't even have to go into the f equals M A for object B and that's it for this one. So hopefully that made sense and thanks for watching.

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