The gravitational field in a region is given by \(\mathbf{E}=(5 \hat{\mathbf{i}}+12 \hat{\mathbf{j}})…
The gravitational field in a region is given by \(\mathbf{E}=(5 \hat{\mathbf{i}}+12 \hat{\mathbf{j}}) \mathrm{Nkg}^{-1}\). If a particle of mass \(2 \mathrm{~kg}\) is moved from the origin to the point \((12 \mathrm{~m}\), \(15 \mathrm{~m}\) ) in this region, the change in gravitational potential energy is
\(-450 J\)
\(-480 J\)
\(-240 J\)
\(-500 J\)
Solution
Given, \(\mathbf{E}=(5 \hat{\mathbf{i}}+12 \hat{\mathbf{j}}) \mathrm{Nkg}^{-1}, m=2 \mathrm{~kg}\) and \(\mathbf{r}=(12 \hat{\mathbf{i}}+15 \hat{\mathbf{j}}) \mathrm{m}\)
Gravitational potential,
\(\begin{aligned}
V & =-\int_0^r \mathbf{E} \cdot d \mathbf{r} \\
& =-\int_0^r\left(E_x \hat{\mathbf{i}}+E_y \hat{\mathbf{j}}\right)\left(d r_x \hat{\mathbf{i}}+d r_y \hat{\mathbf{j}}\right) \\
V & =-\left[\int_0^x E_x d r_x+\int_0^y E_y d r_y\right] \\
V & =-\left[5 \int_0^{12} d r_x+12 \int_0^{15} d r_y\right] \\
V & =-[5(12-0)+12(15-0)] \\
V & =-240 \mathrm{~J} \mathrm{~kg}^{-1}
\end{aligned}\)
Hence, the change in gravitational potential energy of particle of \(2 \mathrm{~kg}\) mass,
\(E=m V=-2 \times 240=-480 \mathrm{~J}\)
So, the correct option is (b).