Question

The gravitational field in a region is given by the equation $\overrightarrow{E}=(5\hat{i}+12\hat{j})\text{N/kg}$. If a particle of mass $2\text{kg}$ is moved from the origin to the point $(12\text{m},5\text{m})$ in this region, the change in the gravitational potential energy is

• A. $-480\text{J}$
• B. $480\text{J}$
• C. $-240\text{J}$
• D. $240\text{J}$

Answer 1

The correct option is C $-240\text{J}$

The change in potential is going from the initial position to the final position in a gravitational field $E$ is given by

$V_f-V_i=-{\int }_i^f\overrightarrow{E}\cdot \overrightarrow{dr}$

Since $\overrightarrow{E}$ is constant

$\therefore V_f-V_i=-\overrightarrow{E}\cdot {\int }_i^f\overrightarrow{dr}$

$\Rightarrow \Delta V=-\overrightarrow{E}\cdot \overrightarrow{r}...........\left(1\right)$

Given:
$\overrightarrow{E}=(5\hat{i}+12\hat{j})\text{N/kg}$

and position vector will be

$\overrightarrow{r}=(12-0)\hat{i}+(5-0)\hat{j}=(12\hat{i}+5\hat{j})\text{m}$

So, eq. $\left(1\right)$ becomes

$\Delta V=-(5\hat{i}+12\hat{j})\cdot (12\hat{i}+5\hat{j})$

$\Rightarrow \Delta V=-(60+60)=-120\text{J/kg}$

The change in potential energy is given by

$\Delta U=m\Delta V=2\times (-120)=-240\text{J}$

Hence, $\left(c\right)$ is the correct choice.

Why this question ? Note- If the gravitational field is constant in both magnitude and direction, the change in potential can be directly calculated as $V=-\overrightarrow{E}\cdot \overrightarrow{r}$