Fundamental Theorem of Line Integration

[Gee Wiz]

Homework Statement

Suppose that F is the inverse square force field below, where c is a constant.
F(r) = c*r/(|r|)^3
r = x i + y j + z k
(a) 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 these points to the origin.

Homework Equations

The Attempt at a Solution

Well this is a conservative force because it is dealing with gravity. So i know that the solution is going to be something like F(d2-d1). But how do i write that out? I think that |r| just equals d1 or d2 depending on which one is selected. But how do i get r?

 

[Gee Wiz]

I also thought something like (c*P2)/(d2)^3-(c*P1)/(d1)^3..but that's not right, so i must be missing something

 

[Dick]

I also thought something like (c*P2)/(d2)^3-(c*P1)/(d1)^3..but that's not right, so i must be missing something

Try and guess a potential function that gives you that vector field as a gradient. If you know something about gravity, you might already know the form of the potential function.

 

[Gee Wiz]

So apparently I know nothing about gravity because i can't guess the potential function. I would think that all the vectors would point in towards the more massive object. (in this case towards the origin)

 

[Dick]

So apparently I know nothing about gravity because i can't guess the potential function. I would think that all the vectors would point in towards the more massive object. (in this case towards the origin)

Which direction they point depends on the sign of c. Try computing the gradient of 1/|r|. Does it look anything like your vector field?

 

[Gee Wiz]

since r(t)=xi+yj+zk isn't lrl=sqrt(3). That isn't the gradient is it..because del f is like (fx,fy,fz,..etc) ...

 

[Dick]

since r(t)=xi+yj+zk isn't lrl=sqrt(3). That isn't the gradient is it..because del f is like (fx,fy,fz,..etc) ...

|r|=sqrt(x^2+y^2+z^2). How is that sqrt(3)?

 

[Gee Wiz]

sorry i meant lr'(t)l