Question

Force acting on a particle in a conservative force field is:
(i) $F=(2\hat{i}+3\hat{j})$
(ii) $F=(2x\hat{i}+2y\hat{j})$
(iii) $F=(y\hat{i}+x\hat{j})$
Find the potential energy function, if it is zero at origin.

Answer 1

$U=-\int F_{x}dx-\int F_{y}dy$

$\left(i\right)$

$U=-\int 2dx-\int 3dy=-2x-3y$

$\left(ii\right)$

$U=-\int 2xdx-\int 2ydy=-x^2-y^2$

$\left(iii\right)$

$U=-\int ydx-\int xdy=-\int d\left(xy\right)$

Since, $d\left(xy\right)=xdy+ydx$

$⟹U=-xy$