The gravitational field in a region is given by $\vec{E} = (5\,N/kg)\,\hat{i} + (12\,N/kg)\,\hat{j}$. If the potential at the origin is taken to be zero,then the ratio of the potential at the points $(12\,m, 0)$ and $(0, 5\,m)$ is:

$A$ sphere of mass $M$ and radius $R_2$ has a concentric cavity of radius $R_1$ as shown in the figure. The force $F$ exerted by the sphere on a particle of mass $m$ located at a distance $r$ from the center of the sphere varies as $(0 \le r \le \infty)$. Which of the following graphs correctly represents this variation?

At what distance from the centre of the moon is the point at which the strength of the resultant gravitational field of the Earth and the Moon is equal to zero (in $,R$)? The Earth's mass is $81$ times that of the Moon,and the distance between the centres of these bodies is $60\,R$,where $R$ is the radius of the Earth.

The magnitude of the gravitational field at distances $r_1$ and $r_2$ from the center of a uniform sphere of radius $R$ and mass $M$ are $F_1$ and $F_2$ respectively. The ratio $(F_1 / F_2)$ will be (if $r_1 > R$ and $r_2 < R$):

$A$ uniform sphere has radius $R$ and mass $M$. The magnitude of the gravitational field at distances $r_1$ and $r_2$ from the centre of the sphere are $E_1$ and $E_2$ respectively. The ratio $E_1: E_2$ is ($r_1 > R$ and $r_2 < R$).

The gravitational field in a region is given by $I = (5 \hat{i} + 12 \hat{j}) \text{ N kg}^{-1}$. The change in the gravitational potential energy of an object of mass $3 \text{ kg}$ when it is taken from the origin to a point $(8 \text{ m}, -2 \text{ m})$ is (in $\text{ J}$)