The gravitational field in a certain region is given as
`vec(E) = (3N kg^(-1)) hat(i) + (5N kg^(-1)) hat(j)`
(a) What will be magnitude of gravitational force acting on a particle of mass 3 g which is placed at the origin ?
(b) What will be gravitational potential at point (12m, 0) and (0, 6m) considering that potential at origin is zero ?
(c ) A particle of mass 3 kg is moved from the origin to the point (9m, 4m). Calculate the change of gravitational potential energy.
(d) If the above particle is moved from (5m, 0) to (0, 5m), then find the change in potential energy.

Explanation

`vec(E) = (3)hat(i) + (5)hat(j) = E_(x)hat(i) + E_(y)hat(j)`
(a) As `vec(F) = vec(E).m`
`= 3[3hat(i) + 3hat(j)]`
`= 9hat(i) + 15 hat(j)`
`|vec(F)| = sqrt((9)^(2) + (15)^(2)) = 17.49 N`
(b) As we know `- dV = vec(E).vec(d)r`
At (12m, 0) `V = (3hat(i) + 5hat(j)) . (12 hat(i) + 0hat(j))`
`v = -36J`
At (0, 6m) `V = (3hat(i) + 5hat(j)) . (0 hat(i) + 6hat(j))`
`v = -30J`
(c ) Change of potential `= dV = -int vec(F). vec(dr)`
`= -int vec(E).m vec(dr)`
`= underset((0,0))overset((3,4))int 3(3hat(i) + 5hat(j)).vec(dr) = 3(3hat(i) + 3hat(j)).|vec(r )|_((0,0))^((9,4))`
`= 3(3hat(i) + hat(j)).[xhat(i) + y hat(j)]_((0,0))^((9,4))`
`= 3(3hat(i) + 5hat(j)).[9hat(i) + 4hat(j)]`
`= 3(27 + 20) = 141 J`
(d) On moving the particle from (5m, 0) to (0, 5m), change in potential
`dV = underset((5,0))overset((0,5))int vec(E)m vec(dr) = m underset((5,0))overset((0,5))int (E_(x)hat(i) + E_(y)hat(j))(dx hat(i) + dy hat(j))`
`= m underset((x=0))overset((x=5))int E_(x) dx + m underset((y = 5))overset((y = 0))int E_(x) dy`
`= m underset(5) overset(0)int 3dx + 3 underset(0)overset(y=0)int 5dy`
`= 9(x)^(0) + 15(y)^(5) = 9 xx (-5) +15 xx 5`
= 30 J