A 40m40\operatorname{m} rope hangs freely over a ledge. The density of the rope is 12kg12\operatorname{kg}/mm. If a 5kg5\operatorname{kg} bucket is attached to the end of the rope, how much work is done to pull the rope and the bucket to the ledge?

96040 J ⁡ 96040\operatorname{J}

A 40 m 40\operatorname{m} rope hangs freely over a ledge. The density of the rope is 12 kg 12\operatorname{kg} / m m . If a 5 kg 5\operatorname{kg} bucket is attached to the end of the rope, how much work is done to pull the rope and the bucket to the ledge?

A 40 meter long rope hangs freely over a ledge. The density of the rope is 12 kilograms per meter. If a five kilogram bucket is attached to the end of the rope, how much work is done to pull the rope and the bucket to the ledge? Okay. So this is a lifting problem we're dealing with, and there notice there's a lot happening because we have this rope that's hanging over a ledge. We're pulling up the rope, but we also have a bucket attached to the end. So how am I gonna go about solving this? Well, what I'm gonna do is I'm gonna solve this in steps. So I'm gonna say that we have this ledge right here that we're looking at, and we are pulling this rope over this ledge. So this is gonna be our rope. It's getting pulled up this ledge. And what this is is a 40 meter long rope. Now what we are trying to do is find the work to fully pull the rope to the ledge. So that means pulling it all the way up to this ledge here. Now I know that it also says the bucket is attached to this, but I'm gonna start by finding the work for just the rope, and you'll see why in a moment here. So what I'm gonna do is find the work required for just the rope, which you'll have the w r for, like, work rope, and that's going to be equal to and recall we use this equation for lifting problems. It's the integral from zero to h of my weight per length multiplied by my length function, all integrated with respect to y. This is the equation we use for calculating work. Now the weight per length, that's given to us right here as 12 kilograms per meter. But this is not actually the weight density. This is the mass density since kilograms is a mass. So if I wanna find the weight density, the weight per length rather than the mass per length, what I can do is take this 12 here and multiply it by 9.8 meters per second squared, Earth's gravitational acceleration. 12 times 9.8, well that's gonna give me a 17.6, and now the units for weight per length will be newtons per meter. So at this point I found my weight per length. Now next what I need to do is find my length function. Well, my length function is simply going to be this. It's gonna be my 40 meter long rope, the length of my rope here, 40 meters minus y, because this y here accounts for each portion of distance that this rope gets lifted up, and eventually we make it all the way, meaning this y would be 40 minus 40. That's zero since the rope is all the way at the top. So this would be my length function. So now I can take everything here and plug it into this equation. So our rope is gonna go from zero, and we're pulling it all the way up. So that's gonna go from zero to 40. That's my upper bound there. My weight per length, we figured out, is a 17.6. We calculated this right here. And then my length function is 40 minus y. And then that's all gonna be integrated with respect to y. Now what I can do is take this one seventeen point six and pull it outside of my integral since that's a constant, and then I can take the antiderivative of this portion right here. So the antiderivative of 40 is gonna be 40y, and the antiderivative of y using the power rule is y squared over two, and all of this is gonna be bounded from zero to 40. Now notice how my lower bound is zero, and every single term in here is multiplied by the variable y. That means that everything within these brackets will just go to zero, so I only need to focus on my upper bound of 40. So you need to take 40 and plug it in for y and then multiply that by 117.6 when you calculate this whole result. Doing this should give you 94,080 joules of work, and this is the work required to move the rope all the way up to the ledge of this cliff right here. That is how we can go ahead and find this work. But we are not done with this problem, because we only found the work to move the rope all the way to the ledge. Recall that this problem also tells us a bucket is attached to the end of this rope. How could we find the work required to lift this bucket? Well, what I'm gonna do is now just focus on the bucket itself. And notice something. This bucket is being lifted straight to the ledge. None of the weight of this bucket will change at any point of the lift. The rope is changing because more and more rope is being pulled up, meaning there's gonna be less and less weight dangling over. But this bucket is going to stay the same weight at every point that we lift it. So we can actually say this bucket is gonna have a constant force, meaning that the work required to lift the bucket is simply going to be the equation force times distance, the standard equation when you have a constant force. Now this force is gonna be the weight of the bucket. And we are told that this bucket here has a mass of five kilograms. So if five kilograms is the mass of this bucket, then I need to multiply this by 9.8 meters per second squared, Earth's gravitational acceleration, to get the weight. So all of this right here would be the force of the bucket. And Then that's gonna be multiplied by the distance that it's lifted. Well, how far is the bucket being lifted? We can see it's being lifted 40 meters all the way to the top of the ledge. So 40 meters is gonna be this. So we're going to have five times 9.8 times 40, and all of that is going to come out to 1,960 joules of work. So we now have the work required to lift the bucket, and we have the work required to lift the rope. Now neither one of these are actually the final answer to the problem, because notice we're not asked for just the rope, and we're not asked for just the bucket. We're asked to find the work required to lift these both to the ledge of the cliff. Well, to do that, what I need to do is add the works together. So my total work is going to be equal to the work required to lift the rope plus the work required to lift the bucket. So this is going to be 94,080 joules plus 1,960 joules. Adding these together will give you a total work of 96,040 joules, and that is the total work and the true solution to this problem. So as you can see, all we did to solve this was first calculate the work required to lift up the rope, then the work required to lift the bucket, and then we added those results together to get our final result. So it's pretty straightforward once you actually get these works calculated. The tedious part of these problems is going through the individual process of finding the work to lift each one individually, the rope and then the bucket, and then adding it together. We hope you found this video helpful. That's how you can solve this problem. Let me know if you have any questions, and let's move on.

A 40 m ⁡ 40\operatorname{m} rope hangs freely over a ledge. The density of the rope is 12 kg ⁡ 12\operatorname{kg} / m m . If a 5 kg ⁡ 5\operatorname{kg} bucket is attached to the end of the rope, how much work is done to pull the rope and the bucket to the ledge?

96040 J ⁡ 96040\operatorname{J}

94080 J ⁡ 94080\operatorname{J}

1960 J ⁡ 1960\operatorname{J}

92120 J ⁡ 92120\operatorname{J}

10. Physics Applications of Integrals

Work

Calculus

10. Physics Applications of Integrals

Work

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Multiple Choice

A $40 m$ rope hangs freely over a ledge. The density of the rope is $12 kg$ / $m$ . If a $5 kg$ bucket is attached to the end of the rope, how much work is done to pull the rope and the bucket to the ledge?

$94080 J$

$1960 J$

$96040 J$

$92120 J$

Verified step by step guidance

Step 1: Understand the problem. Work is calculated as the integral of force over distance. Here, the force varies because the rope's weight changes as it is pulled up. Additionally, the bucket's weight remains constant throughout the process.

Step 2: Calculate the weight of the rope per unit length. The density of the rope is given as 12 kg/m. Multiply this by the acceleration due to gravity (g = 9.8 m/s²) to find the force per unit length: $ F_{rope} = 12 \cdot 9.8 $.

Step 3: Set up the integral for the work done to pull the rope. The rope's weight decreases as it is pulled up, so the force at a distance $ x $ from the ledge is $ F_{rope}(x) = 12 \cdot 9.8 \cdot x $. The total work done to pull the rope is $ W_{rope} = \int_{0}^{40} F_{rope}(x) \, dx $.

Step 4: Calculate the work done to pull the bucket. The bucket's weight is constant at 5 kg, so the force is $ F_{bucket} = 5 \cdot 9.8 $. The work done to pull the bucket is $ W_{bucket} = F_{bucket} \cdot 40 $.

Step 5: Add the work done to pull the rope and the bucket together. The total work is $ W_{total} = W_{rope} + W_{bucket} $. Evaluate the integral for $ W_{rope} $ and add it to $ W_{bucket} $ to find the total work.

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