The gravitational field in a region is given by $E=(5 \mathrm{~N} / \mathrm{kg}) \mathrm{i}+(12 \mathrm{~N} / \mathrm{kg}) \hat{\jmath})$. (a) Find the magnitude of the gravitational force acting on a particle of mass $2 \mathrm{~kg}$ placed at the origin. (b) Find the potential at the points $(12 \mathrm{~m}, 0)$ and $(0,5 \mathrm{~m})$ if the potential at the origin is taken to be zero. (c) Find the change in gravitational potential energy if a particle of mass $2 \mathrm{~kg}$ is taken from the origin to the point (12 $\mathrm{m}, 5 \mathrm{~m})$. (d) Find the change in potential energy if the particle is taken from $(12 \mathrm{~m}, 0)$ to $(0,5 \mathrm{~m})$.
$\vec{E}=5 \hat{\imath}+12 \hat{\jmath}$
(a) Force, $\vec{F}=m \vec{E}$
$\vec{F}=2(5 \hat{\imath}+12 \hat{\jmath})$
$\vec{F}=10 \hat{\imath}+24 \hat{\jmath}$
$|F|=\sqrt{(10)^{2}+(24)^{2}}=26 \mathrm{~N}$
(b) $\int d V=-\int E \cdot d r$
$=-\left[\int(5 \hat{\imath}+12 \hat{j}) \cdot(d x \hat{\imath}+d y \hat{j})\right]$
$=-\left[\int 5 \cdot d x+\int 12 \cdot d y\right]$
$V=-(5 x+12 y)+C$
at $(0,0)$, potential $=0$
$0=-[5(0)+12(0)]+C$
$C=0$
So, $V=-5 x-12 y$
At point $(12,0)$
$V=-5 \times 12-12 \times 0=-60 \mathrm{~J} / \mathrm{Kg}$
At point $(0,5)$
$V=-5 \times 0-12 \times 5=-60 \mathrm{~J} / \mathrm{Kg}$
(c) Potential at $(12,5)$
$V=-5 \times 12-12 \times 5=-120 \mathrm{~J} / \mathrm{kg}$
and potential at $(0,0)$ is 0 .
So, change in gravitational potential energy
$\Delta U=m \Delta V$
$=2(-120-0)$
$\Delta U=-240 J$
(d) $\Delta U=m \Delta V$
$=m(-60-(-60))=0$
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