A 75.0-kg wrecking ball hangs from a uniform, heavy-duty chain of mass 26.0 kg. (b) What is the tension at a point three-fourths of the way up from the bottom of the chain?

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Hey, everyone in this problem, we have a heavy steel ball with a mass of 150 kg suspended from a massive strong uniform rope with a mass of 25 kg. We are asked to calculate the tension in the rope in two different cases. OK? So in part one, but when I calculated it at a point 1/5 of the way up from the bottom. And in part two, we wanna do it at a 0.2 3rd from the bottom. All right. So we have four answer choices. In this case, each a different value or the force of tension for part one and part two. And we are gonna come back to these answer choices once we're done working through this problem. So let's go ahead and draw a little diagram of what we go have going on. So we have this heavy steel ball and it is attached to this large rope. Now, we think about the point, OK. If we're at a point and we are looking for the forces acting, we're gonna have some tension force acting upwards, we're gonna have a force of gravity acting down and we're gonna say that up is our positive direction and those are the two forces that we're gonna have to worry about. And the tension is what we are looking for and that's gonna be opposing that force of gravity. So let's start with part one and in part one, we are gonna do this in bull. OK? So we are looking 1/5 up from the bottom. Let's draw a diagram again. Actually, let's use the diagram we've drawn and we're gonna draw some blue on here. OK? So we're gonna look at a point kind of 1/5 of the way up from the bottom. Now, what's important here is that we're told this is a uniform rope. OK? So we have 1/5 of the rope below our point of interest that tells us that 1/5 of the mass of the rope is going to be contributing to the force of gravity. All right. So again, we're looking at a point 1/5 of the way up from our, the bottom of our rope. If we're looking at that point, we have the force of gravity acting from the ball and then we have some force of gravity acting from the portion of the rope that's below where we are at, right? Because this is a uniform rope. If we have 1/5 of the length of the rope below it, then 1/5 of the mass is going to contribute to that force of gravity. All right. So let's think about our summer horses and use that to find the 10 T. OK. So the summer courses in the Y direction is going to be equal to zero because this is suspended, it's not moving in the vertical direction. So that's equal to zero. The forces that we have acting and we have the tension in the positive force or sorry. Yeah, in the positive direction, the force of gravity in the negative direction. So T minus FG is equal to zero, that tells us that the tension is going to be equal to the force of gravity. And we've just talked about this but the force of gravity, we need to consider the force of gravity of the ball itself. Who was the force of gravity of 1/5 of the road recall that the force of gravity is given by the mass multiplied by the acceleration due to gravity. G. We have the mass of the ball multiplied by G plus the ma mass story of 1/5 of the rope multiplied by the acceleration due to gravity G. And we can go ahead and substitute in our values here. So we have that the tension is going to be equal to 150 kg multiplied by 9.8 m per second squared plus 1/5 multiplied by the mass of the rope, which is 25 kg. And again, we're able to do this because this is a uniform rope nice multiplied by the acceleration due to gravity again, 9.8 m per second squared. And if we work all of this out, we get attention t that is equal to 1519 nos. Yeah, let's compare this to our answer choices. So our answer choices for part one, we have a 1019 B 1319 C 1519 or D 1919. So it looks like C is gonna be the correct answer. If you were on a test, you can go ahead and answer that and move on, save yourself some time. But we are gonna work through the second part of this problem as well just to make sure that we've done everything right and to make sure we understand how to do this. OK? All right. So let's go back to our diagram and we're gonna change it for part two and part two says we wanna look two thirds of the way up from the bottom. OK? So now we're gonna be looking way up here and we're gonna have two thirds of the rope below the point of interest. OK? We're gonna do the exact same thing except now, the mass of the rope that contributes to the force of gravity is gonna be the mass of two thirds of the rope instead of 1/5. For part two. Again, we get that the sum of the forces in the Y direction is equal to zero because this is suspended, it's stationary tension minus the force of gravity is equal to zero. OK. The tension is going to be equal to the force of gravity. We know that that's going to be equal to the force of gravity due to the ball plus the force of gravity due to two thirds of the rope. And we can see this is the exact same process force of gravity is the mass multiplied by the acceleration. So we have the mass of the ball multiplied by the acceleration due to gravity G plus the mass of two thirds of the rope multiplied by the acceleration due to gravity G. And substituting in our values, we have 150 kg multiplied by 9.8 m per second squared. OK. So that first part of our equation doesn't change the contribution of the force of gravity from that ball is the same, it's the same ball with the same weight. Now, for the rope, we have two thirds the mass of the rope. OK. So two thirds multiplied by 25 kg multiplied by the acceleration due to gravity 9.8 m per second squared. And if we work this all out, we get that our attention is going to be about 1633.33 newtons. All right. So we expected the correct answer to be B or sorry. C based on our attention in part one, let's take a look at C and compare the tension we found in part two. So for option C we had 1519 Newtons for part one and 1633.3 Newtons for part two. So that is indeed the correct answer. Thanks everyone for watching. I hope this video helped see you in the next one.

A 75.0-kg wrecking ball hangs from a uniform, heavy-duty chain of mass 26.0 kg. (b) What is the tension at a point three-fourths of the way up from the bottom of the chain?

Verified Solution

Key Concepts

Tension in a Rope or Chain

Weight and Gravitational Force

Static Equilibrium