Question

Gravitational field in $x-y$ plane is given as $\overrightarrow{g}=(2x\hat{i}+3y^2\hat{j})\text{N/kg}$. The difference in gravitational potential between $2$ points $\text{A}(2,4)$ & $\text{B}(6,0)$ is

• A. $V_A-V_B=32\text{J/kg}$
• B. $V_A-V_B=64\text{J/kg}$
• C. $V_A-V_B=-64\text{J/kg}$
• D. $V_A-V_B=-32\text{J/kg}$

Answer 1

The correct option is D $V_A-V_B=-32\text{J/kg}$

Given gravitational field is

$\overrightarrow{g}=(2x\hat{i}+3y^2\hat{j})\text{N/kg}$

Relation between gravitational field and gravitational potential is

$-dV=\overrightarrow{g}\cdot \overrightarrow{dr}$

$\Rightarrow -dV=(2x\hat{i}+3y^2\hat{j}).(dx\hat{i}+dy\hat{j})$

$-dV=2xdx+3y^{2}dy$

Integrate both sides from point $\text{A}(2,4)$ to point $\text{B}(6,0)$

$\therefore \underset{A}{\overset{B}{\int }}-dV=2\underset{2}{\overset{6}{\int }}xdx+3\underset{4}{\overset{0}{\int }}{y}^{2}dy$

$\Rightarrow V_A-V_B=2{\left[\frac{x^2}{2}\right]}_2^6+3{\left[\frac{y^3}{3}\right]}_4^0$

$\Rightarrow V_A-V_B=\frac{2}{2}[36-4]+\frac{3}{3}[0-64]$

$\therefore V_A-V_B=-32\text{J/kg}$

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