Question

Calculate the work done by a force F vector = (x-y) i^ +xy j^

in moving a particle

counterclockwise along the circle x² +y² =4 from the point (2,0) to the point

(0, -2).

Answer 1

$W=\int_C\vec Fd\vec l=\int_CF_xdx+F_ydy,$

$F_x=x-y,~F_y=xy,$

$x=2\cos t,~y=2\sin t,$

$dx=-2\sin t,~dy=2\cos t,$

$t_1=0,~t_2=\frac{3\pi}2,$

$W=\int_0^{\frac{3\pi}2}((2\cos t-2\sin t)(-2\sin t)+(2\sin 2t)(2\cos t))dt=\frac 23+3\pi.$