Power of a Winch on an Incline

by Patrick Ford
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Hey guys, so let's check this one out together. Here we have a block that is 1400 kg 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, that's gonna pull this block up the incline. So this is probably happening like a rock quarry or something like that. I don't know. So I'm gonna 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 delta V. Over delta T. But it's also equal to the work that is done, divided by delta T. Remember that power that work and changing energy means the same thing. So we're actually gonna do here is because we're trying to figure out the force of the cable. We're actually gonna use this work equation. Now, what I have is I actually have the delta T. And that's just gonna be the 25 seconds. That's this delta T. 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 gonna call this some D. Here times the cosine of the angle between those two vectors, right? So this tea and this d. They both pointed the same direction up the incline, Therefore the angle between them is zero and this just equals one. And now we're just going to buy this by delta T. So in order to figure that 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 gonna go ahead and list all the things I know about this problem 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 zero. I'm also told that the coefficient of kinetic friction is mu k 0.4. And I have that this delta t. 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 R. D. This is I'm gonna call this H is equal to 60.2 m. 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, it's really just gonna 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 thing has pulled at constant velocity is because I have an MG. But this MG has a component down the incline. This is MG. X. And I also have a friction force. So this is gonna be my friction K. That is preventing this thing right? That's actually exerting a force downward again because this thing is being pulled up at constant speed this way. So this is gonna be my V. So if I go ahead and set up in Newton's laws I need to set an F. Equals M. A. In order to figure out this tension here. So I'm gonna have to do that. So F. Equals M. A. Now remember what we know is that the acceleration is equal to zero, it's constant speed. So we actually know that this is just gonna be zero. So this allows me to set up an equation relating on my forces. I'm gonna pick the upward direction to be positive because that's where the block is going. So therefore my attention is gonna be positive and this is gonna be minus MG X minus friction kinetic equals zero. When you move both of these terms over to the other side they built become positive and I'm just gonna go ahead and expand them. Remember that MG. X. Is just equal to MG. Times the sine of data. Remember that friction is on an inclined plane, it's mu K. Times the normal force and the normal force is MG times the cosine. Theta. 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 feta, we also have the coefficient. So I'm just gonna have to put this in. Unfortunately this is gonna be really long. But you can just plug it into your calculator and follow it all. Follow along. So the mass is gonna be 1400 G. Is 9.8. Then we have the sine of 37 plus 0. 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 expression is 12 6 40 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 is because I don't have what that devalue is. So let's go ahead and do that. So what I'm gonna do here is I'm gonna 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 what I need, I need that distance. And I also know what the angle of the incline is 37°. And I'm also told with 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 gonna have to use a sine function on 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 with 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 sine theta. So this equals the height which is 60 points two divided by the sine of 37. If you go ahead and work this out, what you're gonna get is a distance of exactly 100 m. 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 gonna be 12 6 40 times the distance, which is 100 then we're gonna divided by the 25 seconds it takes. And when you go ahead and do that, you're gonna get exactly 50,560 watts. That's how much energy this is. Uh this has to actually, 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 kW. That's another way you might see that written. Alright guys, so that's it for this one.