Question

1.7. Consider the vector field F(x,) = (ve" - 1 + xey a. Show that is a conservative force b. Find the potential function for F c. Find the work performed by the force field F on a particle that moves along the saw-tooth curve represented by the parametric equations: x = 1+ sin(sint) and y = (2/) sin(sin), with OSIS8r.

Could someone please help me with this problem? I would really appreciate it.

Answer 1

To show that a given force field is conservative, to find the potential function of the force field and then to calculate the work performed by the force in moving a particle along a given finite path.

17 a. F (x, y) = (y ezy_l) txe z 3 Fx Fal ļ ã Å Tyear, xey o I Î (o-o) -i (0-0) + Ê ( + xyet-epysy) 2 o î - o ĝ to ĥ z ő As Ž x řao, . F is conservative. d. Let y be the potential function. on yeay - i) ît ze ng 7 to k 220 ↑ +243 + 28 Ŕ ..2% 2 yeay! (from w)) Integrating with respect to x, assuming y to be a constant , szy ey_x+ f(y) ety-x+ f(y) consi -- (2) ant any is assumed constant Also, 20 x exy Integrating with respect to my assuming to be a constant, &zxo en + g(x) z e xy + g(x) (3)

- Comparing (2) and (3), 6(y) = C and g(x) 2-2 , ( being 62 a e xy - xt fly constant independent of both x and y: 2 exy x tc whire C is a constant - independent of x andy. .. the potential function is, e x te. c. Initial point = (10 + sin" (rino)) , ( 2 sin "Erino). -([0+ aim to), 3 25 409) (Pelony 2(10+ 0), (*) 7 2 6T Final point = ( (8T + sint (sin 8)), 27 sin" (sin(8))} ( Putting t287) . ( (8 + (ain "(O))), (2 sin" (o 28T+0), Oxo)) ? (8 ; 0). : Work performed by the force field F 2 l F. dr z I o. dr a (dø d -&initial final final initial H over the saw tooth artooth www initial over the saw tooth initial over the saw tooth - CUC 2 0 (81, 0) 8 (0,0) Path 2 (ne 8TXO _ 8. + c) -(eoxo - 0 +0 )

280-85 +&- goto-€ .2 -8 π unit .