$A$ solid sphere of mass $M$ and radius $a$ is surrounded by a uniform concentric spherical shell of thickness $2a$ and mass $2M$. The gravitational field at distance $3a$ from the centre will be

The mass density of a planet of radius $R$ varies with the distance $r$ from its centre as $\rho(r) = \rho_{0} \left(1 - \frac{r^{2}}{R^{2}}\right)$. Then the gravitational field is maximum at:

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 mass of the Earth is $81$ times the mass of the Moon and the distance between their centres is $R$. The distance from the centre of the Earth where the gravitational force will be zero is

Two identical heavy spheres are separated by a distance $10$ times their radius. Will an object placed at the mid-point of the line joining their centres be in stable equilibrium or unstable equilibrium? Give reason for your answer.

Two masses $90 \ kg$ and $160 \ kg$ are separated by a distance of $5 \ m$. The magnitude of the intensity of the gravitational field at a point which is at a distance $3 \ m$ from the $90 \ kg$ mass and $4 \ m$ from the $160 \ kg$ mass is (Universal gravitational constant,$G=6.67 \times 10^{-11} \ N \ m^2 \ kg^{-2}$)