The distance of the planet Pluto from the Sun is $40$ times the distance of the Earth from the Sun. Assuming the masses of the Earth and Pluto are equal,find the ratio of the gravitational force exerted by the Sun on the Earth to that on Pluto.

Find the acceleration of our galaxy due to the nearest comparably sized galaxy. The approximate mass of each galaxy is $8 \times 10^{11}$ solar masses,and they are separated by $2 \times 10^6$ light-years. Each galaxy has a diameter of $10^5$ light-years. (Assume $1 \text{ light-year} \approx 10^{16} \text{ m}$,gravitational constant $G \approx 10^{-10} \text{ Nm}^2/\text{kg}^2$,and mass of the Sun $= 2.0 \times 10^{30} \text{ kg}$)

If two identical spherical bodies of same material and dimensions are kept in contact,the gravitational force between them is proportional to $R^{x}$,where $x$ is a non-zero integer. [Given: $R$ is the radius of each spherical body]

Two concentric shells of uniform density of mass $M_1$ and $M_2$ are situated as shown in the figure. The forces experienced by a particle of mass $m$ when placed at positions $A, B$ and $C$ respectively are (given $OA = p, OB = q$ and $OC = r$):

Two spheres of radius $R$ are placed in contact as shown in the figure. If the mass density of each sphere is $\rho$,find the gravitational force between them.

$A$ straight rod of length $L$ extends from $x = a$ to $x = L + a$. The gravitational force it exerts on a point mass $m$ at $x = 0$,if the mass per unit length of the rod is $A + Bx^2$,is given by