Question

show that \[\vec{F}=k(x\hat{i}+y\hat{j})\] does zero work on a particle that moves once uniformly counterclockwise around the unit circle in the \(xy\)-plane

Answer 1

I'll post what I did (that got zero) let me know if this is the way to do it...

Answer 2

I parametrized with \(x=\cos t,\;y=\sin t\) so \[\vec{F}=k\left\langle \cos t,\;\sin t \right\rangle\]and \[\vec{r}=\left\langle \cos t,\;\sin t \right\rangle\]\[d\vec{r}=\left\langle -\sin t,\;\cos t \right\rangle dt\]So\[W=\int_C \vec{F}\cdot d\vec{r}=\int^{2\pi}_0 \left\langle \cos t,\;\sin t \right\rangle \cdot \left\langle -\sin t,\;\cos t \right\rangle dt =\int^{2\pi}_0 0 \,dt=0 \]