The gravitational field in a region is given by $\mathbf{I}=(5 \hat{\mathbf{i}}+12 \hat{\mathbf{j}})…
The gravitational field in a region is given by $\mathbf{I}=(5 \hat{\mathbf{i}}+12 \hat{\mathbf{j}}) \mathrm{Nkg}^{-1}$. The change in the gravitational potential energy of an object of mass $3 \mathrm{~kg}$ when it is taken from the origin to a point $(8 \mathrm{~m},-2 \mathrm{~m})$ is
$1 \mathrm{~J}$
$16 \mathrm{~J}$
$48 \mathrm{~J}$
$3 \mathrm{~J}$
Solution
Given, gravitational field intensity,
$\mathbf{I}=(5 \hat{\mathbf{i}}+12 \hat{\mathbf{j}}) \mathrm{Nkg}^{-1}$
Mass of object, $m=3 \mathrm{~kg}$
We know that, change in potential energy from to $P(8 m,-2 m)$ is given by
$\Delta U=m \Delta V=m \int_{(0,0)}^{(8,-2)} \mathbf{I} \cdot d \mathbf{r}$
$=3 \int_{(0,0)}^{(8,-2)}(5 \hat{\mathbf{i}}+12 \hat{\mathbf{j}}) \cdot(d x \hat{\mathbf{i}}+d y \hat{\mathbf{j}})=3[5 x+12 y]_{(0,0)}^{(8,-2)}$
$\Delta U=3[40-24]=48 \mathrm{~J}$