Winding a chain A 30-m-long chain hangs vertically from a cyli...

Question

Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.

a. How much work is required to wind the entire chain onto the cylinder using the winch?

Accepted Answer

Welcome back, everyone. In this problem, a 40 m long cable hangs vertically from a platform. The cable has a linear mass density of 4 kg per meter. How much work is required to wind the entire cable onto the platform, assuming the acceleration due to gravity is 9.8 m per second squared and there is no friction. A says it's 31,360 joules. B 39,200 joules, C 45,460 joules, and D 78,400 joules. Now let's try to visualize our problem to help us solve, OK? So we know that we have a platform. And we have a rope, OK, or a long cable that's hanging off of our platform. OK, so let's say that this is our cable here. And you kind of cross it, right? And now, if we think about it, if we're thinking about an infinitesimal piece of cable, then we can imagine that we've already wounded some of the cable up on the platform with a length of X. And let me just, let me show you a bit of my drawing skills here. Let's also put that and show it looking like a cable. That looks a little bit better. So if we, if we've wound up 10 m of cable already on the platform, then that means. That we will have 40 minus X meters of cable remaining, OK, because the cable is 40 m long. And now in that case this tells us then that the weight pulling down on an infinitesimal piece of cable clinging to the platform is going to be equal to its mass multiplied by gravity. OK, the acceleration due to gravity, and since it has a linear mass density of 4 kg per meter, then its mass is going to be equal to 4 multiplied by 40 minus X. In other words, the linear mass density multiplied by the length of cable that's remaining, and if that's the case, then its weight is going to be equal to 4 multiplied by 40 minus X multiplied by G. So if we think about it here, then an equal amount of force must be applied on the infinite decimal piece to lift it and to wind the infinite decimal piece of cable onto the platform, it only needs to move an infinite decimal amount D X. So here we can say that the corresponding work. Corresponding work. OK, to move. An infinitesimal amount D X of the cable, OK. Is going to be DW. OK. A small piece of work, which is gonna be 4 multiplied by 40 minus X multiplied by GD X. Because we're moving DX and among DX. I know that means the work to find the work to lift the entire cable we would integrate with respect to W on the left, which would give us W and with respect to X on the right. So it's going to be the integral between 0 and 40 or 4 multiplied by 40 minus XG with respect to X. This is the work required or the total work required to move or or wind the entire cable onto the platform. Now if we think about it, G is a constant, so you can take it out of our definite integral. And that will leave us with the integral of 40 minus X with respect to X. I know when we integrate we're going to have 4G multiplied by 40 X minus 1/2 of X squared between the bones of 0 and 40. Now, in that case, we substitute our value for a G 9.8 m per second squared, OK. So that's 4 multiplied by 9.8, multiplied by 4 multiplied by 40 minus 40 square divided by 2. And we know if we substitute our lower limit, we know that that would be equal to 0. So now, in this case, this is gonna be 4 multiplied by 9.8, multiplied by 4 multiplied by 40, 1600 minus 40 squared, that's 1600. 40 multiplied by 40. My apologies. That's 1600 minus 40 square divided by 2, which is 800. So this is 4 multiplied by 9.8 multiplied by 800, which equals 31,360 joules thus. The total work required is 31,360 joules. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.
a. How much work is required to wind the entire chain onto the cylinder using the winch?

Key Concepts

Work Done by a Variable Force

Linear Density and Weight Calculation

Setting up and Evaluating Definite Integrals

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