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Fundamental Theorem of Line Integral question

  • Thread starter nfljets
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The question is suppose that F is an inverse square force field, that is,

F(r)=cr/|r^3|

where c is some constant. r = xi + yj + zk. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from those points to the origin.

I'm not exactly sure how to do this or even where to start?

Usually F denotes a vector field, but in case it doesn't...?

Any help would be greatly appreciated!
 

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  • #2
LCKurtz
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The question is suppose that F is an inverse square force field, that is,

F(r)=cr/|r^3|

where c is some constant. r = xi + yj + zk. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from those points to the origin.

I'm not exactly sure how to do this or even where to start?

Usually F denotes a vector field, but in case it doesn't...?
Your || signs should be around just the r, but Why do you say it isn't a vector field? You are given r = <x, y, z> and

[tex]\vec F = \frac {c}{|\vec r|^3}\ \vec r =
\langle \frac{cx}{(x^2+y^2+z^2)^{\frac 3 2}},\frac{cy}{(x^2+y^2+z^2)^{\frac 3 2}},\frac{cz}{(x^2+y^2+z^2)^{\frac 3 2}}\rangle[/tex]

What does the fund. theroem of line integrals have to say about this?
 
  • #3
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I am a bit rusty on vector fields, but I bielive the situation is as follows:
(1) Notice that in every point r=(x,y,z) in space the field F(r) is a vector along the ray coming from the center of coordinate through this point. So you can picture the entire field like a blast from singe central point 0.
(1) Think of two points A, B located at the same distance from 0 - that is, on a sphere with radius |A| - and a path, also on this sphere, connecting them. To move a particle along this path would cost you no energy since you are always orthogonal to force direction.

(2) Conclude that for all it matters, the points P1 and P2 can be located on the same ray coming from the origin (while preserving only their distances from it), and the work accounted for is the work along the path from distance |P2| to distance |P1| along this ray.

To do this formally, look for the term "conservative field", for which the work between two points is path-independent. Prove that your field is conservative, and apply the argument above.
 
  • #4
LCKurtz
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I am a bit rusty on vector fields, but I bielive the situation is as follows:
(1) Notice that in every point r=(x,y,z) in space the field F(r) is a vector along the ray coming from the center of coordinate through this point. So you can picture the entire field like a blast from singe central point 0.
(1) Think of two points A, B located at the same distance from 0 - that is, on a sphere with radius |A| - and a path, also on this sphere, connecting them. To move a particle along this path would cost you no energy since you are always orthogonal to force direction.

(2) Conclude that for all it matters, the points P1 and P2 can be located on the same ray coming from the origin (while preserving only their distances from it), and the work accounted for is the work along the path from distance |P2| to distance |P1| along this ray.

To do this formally, look for the term "conservative field", for which the work between two points is path-independent. Prove that your field is conservative, and apply the argument above.
So, what's your question? He has given you intuitive reasoning and told you what to do.
 

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