If the distance between the centres of Earth and Moon is $D$ and the mass of Earth is $81$ times that of the Moon,at what distance from the centre of Earth will the gravitational field be zero?

$A$ spherical shell is cut into two pieces along a plane as shown in the figure. $P$ is a point on the plane of the cut. The gravitational field at $P$ due to the upper part is $I_1$ and that due to the lower part is $I_2$. What is the relation between them?

Six objects are placed at the vertices of a regular hexagon. The geometric centre of the hexagon is at the origin with objects $1$ and $4$ on the $X$-axis (see figure). The mass of the $k$-th object is $m_k = k^i M |\cos \theta_k|$, where $i$ is an integer, $M$ is a constant with the dimension of mass, and $\theta_k$ is the angular position of the $k$-th vertex measured from the positive $X$-axis in the counter-clockwise sense. If the net gravitational force on a body at the centroid vanishes, the value of $i$ is

Gauss's law for gravitation is given by:

Consider two solid spheres of radii $R_{1} = 1 \; m$ and $R_{2} = 2 \; m$ and masses $M_{1}$ and $M_{2}$,respectively. The gravitational field due to sphere $(1)$ and $(2)$ are shown in the graph. The value of $\frac{M_{1}}{M_{2}}$ is

The gravitational potential versus distance $r$ graph is represented in the figure. The magnitude of the gravitational field intensity is equal to....... $N/kg$.