Hey, guys. So let's check this one out together here. We have a block that is 1400 kilograms that gets pulled up by a winch and cable, which is basically just like a cable that's attached to a spinning or cranking motor. It's going to pull this block up the incline, so this would probably happen in like a rock quarry or something like that. I don't know. So, we're going to call this force of the cable the tension, right, that's tension by a cable. What I want to ultimately figure out is the power that is due to the force of this cable. So how do we figure this out here? Remember that the power equation is just ΔE over Δt, but it's also equal to the work that is done divided by Δt. Remember that power - sorry, that work and change in energy mean the same thing. So what we're actually going to do here is because we're trying to figure out the force of the cable, we're actually going to use this work equation. Now what I have is I actually have the Δt and that's just going to be the 25 seconds. That's this Δt over here. And remember that work is always equal to force times distance. So what I can do here is I can set this up as the tension force times the distance that this block gets pulled up, I'm going to call this some d here, times the cosine of the angle between those two vectors. Right? So this t and this d. They both point in the same direction up the incline, therefore, the angle between them is 0 and this just equals 1. And now we're just going to divide this by Δt. So in order to figure out the power, I'm gonna actually need to figure out the tension force. I'm gonna actually have a number for it and I also need to figure out the distance that this block gets pulled up. I don't have either of those numbers. So how do we do this? First, I'm actually going to go ahead and list all the things I know about this problem. I already have the mass, I know that the incline angle is 37 degrees, and I know that this block is pulled up at constant speed. What that means is that the acceleration is equal to 0. I'm also told what the coefficient of kinetic friction is, μk is 0.4, and I have that this Δt here is 25 seconds. The only other thing that I know here is that the height of the incline, which is this over here not our d. This is I'm gonna call this h is equal to 60.2 meters. So let's go ahead and get started here. The first thing I want to do is figure out the tension that's in the cable. And because I know a couple of things about the forces in this problem, this is really just going to turn into a forces on an inclined plane problem. So remember that there's this force that's pulling this up, but the reason that this is pulled at constant velocity is because I have an mg, but this mg has a component down the incline, this is mgx. And I also have a friction force, so this is going to be my friction k that is preventing this thing, right? It's actually exerting a force downwards again because this thing is being pulled up at constant speed this way. So this is going to be my V. So if I go ahead and sort up in Newton's laws, I need to set up an F = ma in order to figure out this tension here. So, I'm going to have to do that. So F = ma. Now remember, what we know is that the acceleration is equal to 0, it's a constant speed, so we actually know that this is just going to be 0. So this allows me to set up an equation relating all my forces. I'm going to pick the upward direction to be positive because that's where the block is going. So therefore my tension is going to be positive, and this is going to be minus mgx minus friction kinetic equals 0. When you move both of these terms over to the other side, they both become positive, and I'm just going to go ahead and expand them. Remember that mgx is just equal to mg times the sine of θ, and remember that friction is on an inclined plane. It's μk times the normal force and the normal force is mg times the cosine of θ. We've seen this written a bunch of times, so hopefully that's familiar to you. So if I'm sorry, this is actually supposed to be positive as well. Now if you go through these variables, we actually have all of them. We have mass, we have g, we have the θ, we also have the coefficient. So I'm just going to have to plug this in. Unfortunately, this is going to be really long, but you can just plug it into your calculator and follow it along. So the mass is going to be 1400, g is 9.8, then we have the sine of 37 plus 0.4 times 1400 times 9.8 times the cosine of 37, just make sure your calculator is in degrees mode, and what you should get when you plug both of these things in and sort of add them together or you can just plug it in as long as expression is 12640 newtons. So that's the first variable I need. I need the tension. So that's done. The next thing I need is I need to figure out what the distance is along the cable because I don't have what that d value is. So let's go ahead and do that. What I'm going to do here is I'm going to draw a simplified version of the triangle because there's a lot going on here. So what's happening here is I've got this triangle like this and I've got some of the values. I've got the d, which is what I actually need, what I need, I need that distance and I also know what the angle of the incline is, 37 degrees, and I'm also told what h is, 60.2. Now remember for triangles, as long as I have one angle and one side, I can figure out any other piece of the triangle. So if I want to relate to the hypotenuse, which is d, and then the height, which is the opposite side, I'm going to have to use a sine function. I'm a sine function here. So remember that the sine of the angle is related to the opposite side which is h over the hypotenuse. So if I want to figure out what this, with what this distance is, and all I have to do is just trade these two these two variables and switch their places. So, d is equal to h over the sine of θ. So this equals the height which is 60.2 divided by the sine of 37. If you go ahead and work this out, what you're going to get is a distance of exactly 100 meters. So that's how far that this block gets pulled up the incline. So, this d here equals 100. And now we have everything we need to solve this problem here. So we're done with t and d. Now we just plug this all in. So the power is just going to be 12640 times the distance, which is 100, and we're going to divide it by the 25 seconds it takes. And when you go ahead and do that, you're going to get exactly 50,560 watts. That's how much energy this is, this has to this cable actually has to exert, this motor has to exert in order to pull this block up the incline like this. It's a lot of power. So, you know, you might see this written as 50.5 kilowatts. Right? That's another way you might see that written. Alright, guys. So that's it for this one.
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