Question

The gravitational field in a region is given by $\overrightarrow{\mathrm{E}}=\left(5\mathrm{N}{\mathrm{kg}}^{-1}\right)\overrightarrow{i}+\left(12\mathrm{N}{\mathrm{kg}}^{-1}\right)\overrightarrow{j}$. (a) Find the magnitude of the gravitational force acting on a particle of mass 2 kg placed at the origin. (b) Find the potential at the points (12 m, 0) and (0, 5 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 kg is taken from the origin to the point (12 m, 5 m). (d) Find the change in potential energy if the particle is taken from (12 m, 0) to (0, 5 m).

Answer 1

Gravitational field, $\overrightarrow{E}=\left(5\mathrm{N}/\mathrm{kg}\right)\hat{i}+\left(12\mathrm{N}/\mathrm{kg}\right)\hat{j}$
(a) $\overrightarrow{\mathrm{F}}=m\overrightarrow{\mathrm{E}}$
$$\begin{gathered} =2\mathrm{kg}\left[\left(5\mathrm{N}/\mathrm{kg}\right)\hat{i}+\left(12\mathrm{N}/\mathrm{Kg}\right)\hat{j}\right] \\ =\left(10\mathrm{N}\right)\hat{i}+\left(24\mathrm{N}\right)\hat{j} \\ \therefore \left|\overrightarrow{\mathrm{F}}\right|=\sqrt{100+576}=\sqrt{676}=26\mathrm{N} \end{gathered}$$

(b) $V=-\overrightarrow{\mathrm{E}.}\overrightarrow{r}$
Potential at (12 m, 0)$=-60\mathrm{J}/\mathrm{Kg}$

Potential at (0, 5 m) = $-60\mathrm{J}/\mathrm{kg}$

(c)
change in potential=final potential -initial potential
initial potential=potential at the origin=0
final potential=potential at (12,5)
$$\begin{gathered} V=-\overrightarrow{E}.\overrightarrow{r} \\ =-(10\hat{i}+24\stackrel{⏜}{j}).(12\stackrel{⏜}{i}+5\stackrel{⏜}{j}) \\ =-(120+120)\text{J} \\ =-240\text{J} \end{gathered}$$

(d)
$$\begin{gathered} ∆V=-\overrightarrow{E}.∆\overrightarrow{r} \\ ∆\overrightarrow{r}=(12\stackrel{⏜}{i}+0\stackrel{⏜}{j})-(0\stackrel{⏜}{i}+5\stackrel{⏜}{j}) \\ =12\stackrel{⏜}{i}-5\stackrel{⏜}{j} \\ ∆V=-(10\stackrel{⏜}{i}+12\stackrel{⏜}{j}).(12\stackrel{⏜}{i}-5\stackrel{⏜}{j}) \\ =0\text{J} \end{gathered}$$