A box of bananas weighing 40.0 N rests on a horizontal surface. The coefficient of static friction between the box and the surface is 0.40, and the coefficient of kinetic friction is 0.20. (b) What is the magnitude of the friction force if a monkey applies a horizontal force of 6.0 N to the box and the box is initially at rest?

Video transcript

Hey everyone in this problem. We have a 50 Newton animal salt block at rest on a horizontal floor. Okay. The coefficients of static and kinetic friction between the block and the floor are 0.54 and 0.36 respectively A goat pushes on the block, applying a horizontal force, gave magnitude 12 newtons and were asked to determine the magnitude of the friction force that the block is at rest when it's first when that pushes applied. Okay? Alright, so let's just draw this out. We have our horizontal floor, we have our block and then we have a goat here, you can pretend that this picture looks like a goat um is pushing this block. Okay, Alright, so if we draw this as a free body diagram, we have our block in the middle. Okay, We know that there's a normal force pointing up perpendicular to the surface. Okay, that's always there. We have the weight pointing down. Now we have this applied force, Okay? We're gonna call it f from the goat pushing the block and it's going to be pointing to the right, You can have it pointing to the left if you wanted, but according to the little picture we drew, it's gonna be pushing it to the right. Okay, And now the force of friction, the force of friction is going to oppose motion or resist the motion, so it's going to act in the opposite direction. So the force of friction is going to be pointing to the left, we'll call it little F. We're gonna take up into the right to be positive in our coordinate system. Alright, So we want to determine the magnitude of the friction force. Okay? And what we want to do first is look at whether this friction force is going to exceed the threshold, okay. Of static friction. So we're gonna look at what is this maximum static friction force? Is the force applied going to exceed that? Okay. We're told that the block is at rest when the push is applied, but we're not sure if that's going to make the block move after it's pushed or not. Okay, So let's look at F. S max K. The maximum static friction force. And recall that this is going to be equal to mu S. The coefficient of static friction times end the normal force us the coefficient of static friction. We're told in the problem, but we don't know N. Okay. And our normal force N. Is in the Y direction. So let's switch over and look at the sum of the forces in the Y direction. In order to determine now from Newton's second law, we know that some of the forces in the Y direction to the mass times the acceleration in the y direction. The block is not moving up and down. It's just resting on that floor. So the acceleration in the Y direction is zero, which means that the sum of the forces in the y direction is zero. The sum of the forces we have pointing up in the positive Y direction, The normal force N. And pointing down in the negative y direction. We have W. So we get n minus W. Is equal to zero. And so the normal force end, it's going to be equal to the weight W. Of the crate. We're told that the block or sorry, block not the crate. We're told that the block is 50 newtons. So the weight is 50 newtons, which means that our normal force is 50 newtons. Okay, And just watch here, we're given in this problem, the weight of the block, sometimes you'll be given the mass. Okay, so just be careful whether you're given something in newtons or in some unit of weight versus in a unit of mass like kilograms. Alright, so getting back to our maximum static friction force we have in us. The coefficient of static friction which we're told is 0.54. Okay, and then we have our normal force. 50 Newtons which we just found, Which gives us a maximum static friction force of newtons. Yeah. And what we notice is that that F. S max the maximum static friction force of 27 newtons is greater than the applied force F. Of 12 newtons. Okay. If our maximum static friction forces greater than the applied force, this means that the block will not move. Okay, and that static friction applies. So the F that we've drawn in our diagram and our free body diagram is going to be equal to F. S. In this case. Okay, the static friction force. Alright, so let's just add that in. Okay, this is now F. S. Instead of just F. Alright, so we know we're dealing with static friction. What we want to know is what is the magnitude of the friction force. And be careful here. It could be easy to answer that. It's 27 newtons, okay, but that's not the case. 27 Newtons is the maximum static friction force. But recall that when we have static friction, okay, this block is not moving. And so the static friction forces actually just going to oppose the force that we have. Okay, it's not always going to be its maximum value. If it was at its maximum value, we'd have a force pointing to the left of 27 newtons of force pointing to the right of Newton's, that means that the block would be moving to the left, which we know we don't have. Okay, so let's look at this mathematically. Okay, let's use Newton's second law, we know that the sum of the forces in the X direction is equal to the mass times the acceleration in the X direction and because we're not above that threshold, we know that our block is not moving and so our acceleration is zero and the sum of the forces in the X direction is going to be zero. Alright, what are the forces we have in the positive extraction? We have the applied force F. And in the negative extraction we have our force of static friction F. S. We know that F minus F. S. Is equal to zero. Okay? If we give ourselves some more room that tells us that the Force of static friction that we're trying to find going to be equal to the applied force f. Which is 12 newtons. Okay. All right, so like I said, this is gonna be an opposing force. Okay? It's going to be equal to and opposite to the force we're applying in one direction for our static friction force. Okay? It's not always at its maximum value because that would cause the block to move backwards, which is not what friction does. Okay, friction resists or opposes motion does not create it. Alright, so, we're looking at our options, we found that the magnitude of the friction force is going to be b. 12 Newtons. That's it for this one. Thanks everyone for watching. See you in the next video