Hi there all: I have a problem!
I need to find the work done on a particle that moves from $(0,0)$ to a point $(1,1)$ by a strait line $y=x$. The force acting upon the particle is $F = (y , 2x$).
Now how i have attempted the problem is I paramaterized the line $x = y$ to be $r = < t, t >$ by letting $x = t$ which means that $y = t$ too. (I'm hoping that this is how you do it, it's been a while since ive done that maths course)
I know the formula for work is: The integral of $F\cdot dr$ - thus meaning I have to differentiate $r$ to get $dr = < 1, 1>$ then finding the force acting on the particle using $F$ and the particles position of $(x,y)$ to be $(1,1)$ (i.e. i substituted this point into $F$) so $F= (1,2)$.
Then taking the dot product between the two i get $3$. Then taking the integral between $1$ and $0$ I get the answer to be $3$. Now the answer in the textbook says the answer is $1.5$ :( bummer.
Where have I gone wrong? Do i need to paramiterise F differently? or.. i dno ??? :S