Find the work done by the force field F on particle moving along path

[opaquin]

Homework Statement

Find the work done by the force field F on particle moving along path

F(x,y) = -xi + 6yj, <I>C</I>: y = x^3 from (0,0) to (6,216)

The Attempt at a Solution

Drew the graph in the xy plane (y = x^3 is upper limit, x = 0 is lower limit)
W = F ° r'(t) dt
Vector field is not conservative
I want to find r'(t) but am confused as to how, just not sure where to go from here. I have a habit of over thinking problems and am probably doing so. Any assistance is greatly appreciated.

 

[pasmith]

If you're given a path of the form $y = f(x)$, then you can always parametrize it as $(x(t),y(t)) = (t, f(t))$.

 

[opaquin]

in that case, would I just evaluate the line integral?
∫ f(x(t),y(t))||r'||

 

[opaquin]

r(t) = _ i + _ j, _ <= t <= _

r(t) = (t)i + (t^3)j, 0 <= t <= 6
r'(t) = <1, 3t^2)

am I on the right track?

 

[pasmith]

in that case, would I just evaluate the line integral?
∫ f(x(t),y(t))||r'||

You need to calculate
$$\int_0^6 \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,\mathrm{d}t$$

r(t) = _ i + _ j, _ <= t <= _

r(t) = (t)i + (t^3)j, 0 <= t <= 6
r'(t) = <1, 3t^2)

am I on the right track?

Yes.