Help Setting Up Work Over a Path Problem

[erok81]

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?

 

[nonequilibrium]

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

 

[erok81]

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!

 

[nonequilibrium]

Welcome!

 

[erok81]

Turns out, I don't get it.

I have another problem where I am supposed to evaluate $$\int_{C}xydx+(x+y)dy$$ 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)?

 

[vela]

That's the definition of r(t).

 

[erok81]

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.

 

[vela]

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).