Question

1. If the density of dust collected by a single passage of a vacuum cleaner is given by d(x,y) = 5 + xy (k/m), how much dust would be collected by a vacuum cleaner traveling counterclockwise along the upper half of a semi-circle of radius 2 centered at the origin? 2. Find the work done by the force field F = 27 +227 + zkon a particle traveling from (0, 2, -1) to (2,1,1) along a straight line. 3. Are the following vector fields conservative? If so, find its potential function. **+ xy/z+e'a a) F=< zyz+*+2, ***+2, + ze"> b) E=< **** +3*, 5, 202-11/2>

Answer 1

Answer: tool 1-2,0) T2,0) x²+ y² =4 x = 2 coso, y = 2sing DECO,IT) dcq,y) = 5+? y con dlardo. Total dust collected = {G+2²4) de : ? 15+2205²0. 2 sin ordo- -|(5 + 2 fost o 25ôno) v do storsto 8 Go OSPRO) 2 do . f10 do + 16 Aug o sáns do = 10T + 16 asino do iT + 10 (6038/cosodo • LOTT (9) Total dust collected : 10 it (9). (2) Given that ²:21 + x² 5 + ZK dr = dxit dy jt dans

- work done : f § di F. dia a data dy+zdz By Parametridation x=24 | 4=2-7 dx=2dt I dy=-dt 12=26-1 d2=adt t varies from oto F. dr = 2 dx + x²dy + zdz = 2(20+) + Qu'(-2+) + (21-1) (adt) [4-4E+ht-2] at P. dr : [2-27²+ ht] dr .. walk done = f Fidir :${2-4'+ ht] dt 3. 2-4+2 wolk done =