Question

compute the work done by the force F along the curve C. F(x,y) = , C is the circle x^2 + y^2 = 4 oriented counterclockwise.

Answer 1

1)\[A=\oint\limits_{C}\vec{F}\cdot d\vec{l}\quad;\quad\vec{F}=x\cdot\vec{i}-y\cdot\vec{j}=2(\cos{\phi}\cdot\vec{i}-\sin{\phi}\cdot\vec{j})\]
\[d\vec{l}=|d\vec{l}|(-\sin{\phi}\cdot\vec{i}+\cos{\phi}\cdot\vec{j})=2(-\sin{\phi}\cdot\vec{i}+\cos{\phi}\cdot\vec{j})d\phi\]
\[A=\oint\limits_{C}\vec{F}\cdot d\vec{l}=4\int\limits_{0}^{2\pi}(\cos{\phi}\cdot\vec{i}-\sin{\phi}\cdot\vec{j})\cdot (-\sin{\phi}\cdot\vec{i}+\cos{\phi}\cdot\vec{j})d\phi=\]
\[=4\int\limits_{0}^{2\pi}(-cos{\phi}\sin{\phi}-cos{\phi}\sin{\phi})d\phi=-4\int\limits_{0}^{2\pi}\sin{2\phi}\,d\phi=0\]
2) \[A=\oint\limits_{C}\vec{F}\cdot d\vec{l}=\iint\limits_{S}rot\,\,\vec{F}\cdot d\vec{\sigma}\quad;\quad rot\,\,\vec{F}=0\quad\Rightarrow\quad A=0\]