## richyw 3 years ago 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

1. richyw

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

2. richyw

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$