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Preventing a Block From Moving

Patrick Ford
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All right, guys, let's check this one out. We have this 15 kg block that's at rest were given the coefficient of static friction. And we want to figure out how hard you have to push down on this block in order to keep it from moving. So, basically, let's sketch this out. We have a 15 kg block, this on a flat surface, right? And basically what we're gonna be doing is pushing down. I'm gonna call this Force f down. That's what we're trying to find. And we want to push down hard enough so that we can generate enough friction so that a horizontal force which we know is 300 cannot get it moving. All right, So we want to do is we want to, uh, first start off with the free body diagram. So let's go ahead and do that. So we've got our box, we've got RMG that's downwards. And then we also have any applied forces. We know there's two r f down is what we're trying to solve for, and we also have an applied force. It acts to the rights. This is f equals 300. Now there's also going to be some normal force because they're on the floor. And then finally, what happens is without friction. The box would start moving to the right. But if we're going to keep this thing for removing, that's because there has to be some static friction that is opposing this. So there is some friction here that is opposing that, right? So it actually goes. So that actually brings us to the second step, which we've already just talked about. Here. We have to determine whether this friction is static or connected based on the problem text or by looking at all the forces involved. And what we've seen here is that we're pushing down on the block to keep this box from moving basically. So what happens is we know that this friction is going to be static. All right, so let's go ahead and now, right? R f equals m A. So I'm gonna write out my ethical dilemma here. I'm going to just pick a direction of positive up and to the rights now. Usually we would start with the X axis, but because we're looking at a force that's in the Y axis, we're gonna go and start with our Y axis first four, uh, first. So we've got our some of all forces equals m A y. Now this box isn't going up or down in the vertical axis. We know the acceleration is going to be equal to zero. So therefore we expand our forces, remember, our normal is positive minus M g minus f down, and this is equal to zero. Those forces have to cancel. So here's f down. And if I go ahead and just solve for this variable over here by bringing it to the other side and flipping the equation around f down is really just equal to the normal minus mg. Okay, so this f down the force that I need to get this object to prevent this object for moving is going to be equal to the normal minus mg. But the problem is, I actually don't know what this normal force is. So to figure this out whenever I get stuck in one axis, I usually just go to the other access to solve. So I'm gonna go to the X axis forces to now solve for this. So I've got f equals M A and the X axis. So what are our forces? We have our f, which is the 300 minus your F s. And what's the acceleration? Well, here's the kicker, guys, if we're trying to figure out how much we need to push on the block to prevent it from moving, we're basically trying to figure out what is the minimum force that we need so that the static friction exactly balances out the 300. And so what happens is this is gonna be F s, Max. So the minimum force is going to be where the F S Max is going to balance out the 300 if 300 was just a little bit more than it would actually get the object to start moving. So this FS here is actually maximum static friction. And so because of that, because the object doesn't move, then that means that the acceleration has to be equal to zero. These forces have to balance. So that means that your f is equal to your F S max, which is equal to remember mu static times the normal. So remember we came to the X axis because we were stuck and we wanted to figure out what that normal force is. Now we can figure it out. Our normal forces really just going to be equal to your f divided by mu static. So your f is 300 your mu static 0.7, you'll get a normal force of 429. So your normal force is 429. Basically, if the normals 429 then your f S Max is going to be 300 to balance out the 300 that you're pushing it with. So now we have our normal force here, which means that F down is just equal to minus your mg, which is 15 times 9.8. So if you guys go ahead and plug this in your calculators, you're gonna get 282 Newtons. So if you look at our answer choices, that's answer choice, see? All right, guys, let's take this one