Help Setting Up Work Over a Path Problem

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In summary, the student is trying to find the work done by force F from (0,0,0) to (1,1,1). They are using the equation F=3yi+2xj+4zk, but are having trouble getting the r(t) correctly. They are thinking that r(t)= i+j+k and points are used to find the values, but that's not always the case. Finally, they are having trouble understanding what r(t) is and are not sure where to find information on it.
  • #1
erok81
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Homework Statement



Find the work done by force F from (0,0,0) to (1,1,1).

F=3yi+2xj+4zk

The path C_3 U C_4 consisting of the line segment from (0,0,0) to (1,1,0) followed by the segment (1,1,0) to (1,1,1)

Homework Equations



None

The Attempt at a Solution



I am trying to find r(t) for both line segments.

For r(t)_1 I tried (1-0)i+(1-0)k+(0-0)j. I got that from x_2-x_1, etc. This worked out for the first r(t) netting me ti+tj

I then tried this for the second line segment and ended with tk only. The correct r(t) is i+j+tk

So clearly I am solving the r(t) part incorrectly. The book doesn't go over how to get the r(t) - that is where I need the help. I am thinking it might have something to do with the original (0,0,0) to (1,1,1) but I'm not sure.

How does one obtain both r(t)'s when only points are given?
 
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  • #2
Hey, don't worry, your method is basically right:

so your r(t)_1 is completely right and your r(t)_2 makes sense but has a small error: you were thinking of a vector only, correct? When you use the x_2 - x_1, etc... you're determining the direction of the path, that sounds logical, because you're only using the differences (ending point minus start point), but take in account that r(t) is not only a vector pointing in the right direction (your direction is correct!), but is a path that has a beginning and end point! Make a drawing of the x,y,z-axes, the points (0,0,0), (1,1,0) and (1,1,1) and then the arrows pointing from (0,0,0) to (1,1,0) and from (1,1,0) to (1,1,1), you'll see where you went wrong: tk is the path from the origin to (0,0,1), but that's not the path you're taking: it has to go through the point (1,1,0). So how do you "fix" the tk correctly? r(t) = tk + constant, with the constant determining where the path is (the direction is already fixed, of course), and as r(0) = (1,1,0), we get the right answer r(t) = i + j + tk
 
  • #3
Got it. That makes sense. I'll try a few more after work and see if I've got it down.

Thanks for the help!
 
  • #4
Welcome!
 
  • #5
Turns out, I don't get it. :confused:

I have another problem where I am supposed to evaluate [tex]\int_{C}xydx+(x+y)dy[/tex] along the curve y=x^2 from (-1,0) to (2,4).

If I start subtracting the two points I get 3i + 3j. The correct answer for r(t) is ti + t^2j.

I see in the solutions manual they set x=t and therefore get y=t^2...I understand that - although not sure exactly why they used x=t, I am assuming just for ease of solving sake.

So anyway, how did they arrive at xi +yj for r(t)?
 
  • #6
That's the definition of r(t).
 
  • #7
That's where I am having trouble. We just started using r(t) with no explanation of what it is and I can't see anywhere that explains it in the book.

It almost seems like r(t)= i+j+k and then points are used to find the values. But that's only from the problem sets I've done.

I don't really understand what r(t) is so I can't conceptually get it or look it up anywhere outside of my book.
 
  • #8
It's just a vector from the origin to a point on the path, so r(t)=x(t)i + y(t)j + z(t)k. As t varies, the tip of the vector moves along the path.

For example, in your first problem, you had
The path C_3 U C_4 consisting of the line segment from (0,0,0) to (1,1,0) followed by the segment (1,1,0) to (1,1,1).
For C3, you could use r(t) = (1,1,0)t = t i + t j, so as t varied from 0 to 1, the tip would follow the line segment joining (0,0,0) and (1,1,0). For C4, you could use r(t) = (1,1,0)+(0,0,1)t = 1 i + 1 j + t k so as t goes from 0 to 1, r would go from (1,1,0) to (1,1,1) along the line segment connecting them.

In this latest problem, they told you how x and y are related. If you figure out some expression for x, you can just square it to find y. The simplest thing you can do is just set x to the parameter t. Note that this isn't the only parameterization you could use. If you wanted to make things more difficult for yourself, you could, for instance, use x=t2, then y=t4. As long as (x(t),y(t)) follows the curve, you're fine.

By the way, are you sure the problem says from (-1,0) to (2,4) and not (0,0) to (2,4)? The curve y=x2 doesn't pass through (-1,0).
 

1. How do I determine the appropriate path for my work?

The best way to determine the appropriate path for your work is to first understand the task or project at hand. Consider the objectives, resources available, and any potential obstacles. Then, map out a clear and logical path that will help you reach your goal efficiently.

2. What tools or resources can I use to set up my work over a path?

There are several tools and resources that can help you set up your work over a path. Some popular options include project management software, Gantt charts, and mind mapping tools. Additionally, seeking guidance from colleagues or mentors can also be beneficial.

3. How do I prioritize tasks along the path?

Prioritizing tasks along the path involves assessing the importance and urgency of each task. Consider the impact each task has on the overall goal and the timeline for completion. It may also be helpful to break down larger tasks into smaller, more manageable ones.

4. What should I do if I encounter roadblocks or challenges along the path?

It is common to encounter roadblocks or challenges while setting up work over a path. The key is to remain flexible and adaptable. Take a step back and reassess the situation, and then find alternative solutions or workarounds. It may also be helpful to seek advice or assistance from others.

5. How can I ensure that my work is progressing along the path as planned?

Regularly monitoring and tracking your progress is crucial to ensuring that your work is progressing along the path as planned. This can be done through regular check-ins with your team or supervisor, setting milestones and deadlines, and regularly reviewing and adjusting your plan as needed.

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