Hi I am having some trouble with the following:
When I am given some force F and it is in terms of components ie with respect to i, j, k then I have no issue using it to solve line integrals etc, my problem is when it is in the form like
force proportional to distance from Origin, so I know it is $\sqrt{x^2+y^2}$ but now I dont have $i$, $j$ and then I am confused on how to plug in my parametric path and dot with the derivative of the path.
Can someone help to explain it to me,
for example, if we were just asked to find the work done moving a particle on the curve $y=1+x^2$, where the force is proportional to the distance from the origin.
Well I would parametrize $r(t)=(t,1+t^2)$ , $0 \le t \le 1$ and thus $r'(t)=(1,2t)$
but how would I write $F$?
$F=-K\sqrt{x^2+y^2}$
like should I have $F=-K\sqrt{x^2 i + y^2 i } $ or something of that way?
But then how I get $F(r(t))$ ? And how are the components separated so that I can dot it with $r'(t)$ ?
If it were the case I am used to like $F=(x)i+(y)j$ then I would have $F=(t)i+(1+t^2)j$ , etc and dot product would be much more clear, but in the other cases, I am really confused.
So is there even a way I can write it in this form when this is the case? Or do I need to approach it diffirently Please help,
Thanks
For the vectors $$ r = (x,y,z)^t = x i + y j + z k \\ F = (F_x, F_y, F_z)^t $$ the force of a linear spring would be $$ F = -k r = -k \lVert r \rVert \frac{r}{\lVert r \rVert} = -k \lVert r \rVert e_r $$ with $$ \lVert r \rVert^2 = x^2 + y^2 + z^2 $$ and $e_r$ the radial unit vector.