1
$\begingroup$

Given $\vec{F} =(y^2+x^2,−2xy)$. I want to calculate the work done by this force, that is applied on a particle moving from $(0,0)$ to $(3,5)$ in a straight line.

I know that $W=\int_1^2 \vec{F} \cdot d\vec{r}$. The problem is that I don't really now how to properly express $d\vec{r}$ and the integral boundaries.

How do I write this down correctly? Is there some sort of useful parameterization (line integral over continuous vector field)?

$\endgroup$
  • 3
    $\begingroup$ You are correct in that you need to parameterize the line. Hint: a line in with slope $m$ can be parameterized as $(t,mt), t \in \mathbb{R}$. $\endgroup$ – D.B. Feb 28 at 19:06
  • $\begingroup$ I'm new to line integrals and your comment definitely gave me more insight and a very useful hint. Thank you a lot! $\endgroup$ – Zachary Feb 28 at 19:11
1
$\begingroup$

Hint: With $$y=\frac53x$$ we have

$$ W=\int_0^3\left(\left[\frac53x\right]^2+x^2,-2x\left [\frac53x\right]\right)\cdot\left(1,\frac53\right)dx $$

Can you take it from here?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.