A uniform ladder 5.0 m long rests against a frictionless, vert...

Question

A uniform ladder 5.0 m long rests against a frictionless, vertical wall with its lower end 3.0 m from the wall. The ladder weighs 160 N. The coefficient of static friction between the foot of the ladder and the ground is 0.40. A man weighing 740 N climbs slowly up the ladder. Start by drawing a free-body diagram of the ladder. (a) What is the maximum friction force that the ground can exert on the ladder at its lower end?

Accepted Answer

Hey everyone in this problem. We have a painter that places a 13 m long regular ladder against a vertical wall with negligible friction. The Ladder's lower end is five m from the wall. We're gonna take the weight of the ladder to be 250 newtons. And the coefficient of static friction between the floor and the ladders foot to be 0.65. The painter, weighing 650 newtons, climbs gently up the ladder, were asked to determine the maximum friction force that the floor can apply on the ladder's lower end. Okay, so let's just draw this out first. Okay? So we have our ladder in the floor, the wall and our ladder is going to be kind of on an angle like this. Now we're told that this ladder is 13 m long. So the distance here, It's going to be 13 m And we're told that it's five m from the wall, the end. Okay, so this distance here is going to be five m. All right, now, we're thinking about the forces that we have. Okay, we're told that the wall has negligible friction, in case we don't have any friction here, but we will have the normal force n due to the wall. Okay? So N. W and that's gonna be perpendicular to the wall. So it's gonna be pointing to the left. Now, at the bottom of the ladder, we do have friction. Okay, we have friction between the floor and the ladder. Now, if we think about this ladder moving up the ladder is gonna slide, the end is going to slide out to the left, the friction force is going to oppose that. Okay so the friction force, let's draw our forces actually in blue so we can just kind of see them better. These are friction force is going to be acting to the right opposing that motion. So this is going to be friction force. Um Let's call it little F. And now we know that we're stationary, okay, this isn't moving. And we're talking about static friction. So we have F. S. Okay for static friction. What else do we have? Well we're going to have a normal force at the bottom as well. Okay so we're gonna have this normal force due to the floor. We'll call it N. F. And that's going to act perpendicular to the floor. So it's going to act straight up. Okay? And then we're gonna have the weight of the ladder. Okay? This ladder is just a regular ladder, we're gonna take the weight from the middle, the center of gravity, the weight of the ladder. W and we can't forget the person. Okay? So we have a person on our ladder and that is gonna cause another weight w due to the person. All right, we're going to assume that this person is right at the foot of that ladder, they've just started climbing up because that's when we're gonna have more friction there. Okay. All right so we're looking for the maximum friction force and again that's why we put this person right at the bottom because that's when their um wait is gonna be acting more downwards. Um So we're gonna get a larger normal force and a larger friction force. Okay. Alright, so let's pick our coordinate system. The typical up into the right is positive and again, we want to find the maximum friction force. And when we're talking about maximum friction force, recall that we have F. S. Max the maximum static friction force is equal to mu s the coefficient of static friction times end the normal force. Okay. And in this case the friction forces acting at the bottom of the ladder here against the floor. So the normal force that we're concerned about is going to be the normal force acting at that point, which is N. F. All right. So, we know mu s were given that in the problem the coefficient of static friction, but we don't know this normal force N. F. What we do know though, is that this is an equilibrium situation? Okay. We have no net external forces acting. The latter is not moving. And so we can say that the sum of the forces in the X direction is zero. The sum of the forces in the Y direction is zero. Okay. And this ladder is also not rotating. So we can say that the sum of the torques is equal to zero. Okay, So we have these equilibrium conditions. Now we're looking for NF NF x in the Y direction. So let's start with the sum of the forces in the Y direction and see where that takes us. Okay, so some of the forces in the Y direction we know is equal to zero. What forces do we have in the Y direction? Well, acting in the positive Y direction. We have N. F. The normal force the floor and then acting in the negative Y direction. We have the weight W. Of the ladder and we have the weight W. P. Of the painter. And so we get N F minus W minus W. P. Is equal to zero. Okay, now we're trying to solve for an F. Okay, so that we can use it in our equation for F. S max. So N. F is going to be equal to the weight W Plus the weight of the painter P W. P. Sorry, Which is equal to 250 newtons plus 650 newtons. Okay, if we look at the problem, we're given that the weight of the ladder is 250 newtons. And the painter weighs 650 newtons. Okay, so we have 250 plus which gives us a weight We're sorry, a normal force n. f. of 900 newtons. Alright, so now that we have an A and F. We can go back to our Fs max equation. Okay, again, this is what we're looking for. This cowfish or this maximum static friction force and this is equal to mu S and F. The coefficient of static friction were given in, the problem is 0.65. So we have 0.65 times the normal force from the floor, which we just found to be 900 newtons, which gives an F. S. Max of 580 five Nunes. All right, so that maximum static friction force between the floor and the ladder is gonna be 585 newtons. And if we go back up to the answer choices, we see that we get answered. D. Okay, 585 Newtons. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Free-Body Diagram

Static Friction

Torque and Equilibrium

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