Question

Question is given below, please explain it thoroughly?

Answer 1

See below.

Explanation:

a)

`W = int_(r_A)^(r_B) << vec F, vec (dr) >> = int_(r_A)^(r_B) << k(r (vec r)/r), vec (dr) >> = int_(r_A)^(r_B) << k vec r, vec (dr) >> = (k/2 r^2)_(r_A)^(r_B) = 4/2(9^2-(5^2+3^2))=94`

b)

`W = << vec F, vec(Delta s) >> = << F_0 (vec(Delta s))/abs(Delta s),vec(Delta s) >>`

where `vec(Delta s) = vec (p_B)-vec (p_A)` and then

`W =F_0 norm(vec(Delta s))^2/abs(vec(Delta s))=8 xx 5 = 40`

NOTE:

`int_(r(s_1))^(r(s_2))<< vec r, vec (dr)/(ds)>>ds = int_(r(s_1))^(r(s_2))(x,y)cdot(dx/(ds), dy/(ds))ds = int_(r(s_1))^(r(s_2))(x dx/(ds)+y dy/(ds))ds= 1/2int_(r(s_1))^(r(s_2))d/(ds)(x^2+y^2)ds = 1/2 int_(r(s_1))^(r(s_2))(d/(ds)r^2) ds = 1/2 (r(s_2)^2-r(s_1)^2)`