A double star is a system of two stars of masses $\mathrm{m}$ and $2 \mathrm{m}$, rotating about their centre of mass only under their mutual gravitational attraction. If $\mathrm{r}$ is the separation between these two stars then their time period of rotation about their centre of mass will be proportional to :

A solid sphere of uniform density and radius 4 units is located with its centre at the origin $\mathrm{O}$ of coordinates. Two spheres of equal radii 1 unit, with their centres at $\mathrm{A}(-2,0,0)$ and $\mathrm{B}(2,0,0)$ respectively, are taken out of the solid leaving behind spherical cavities as shown in figure. Then :-

A small ball of mass ' $\text{m}$ ' is released at a height ' $\text{R}$ ' above the Earth surface, as shown in the figure. If the maximum depth of the hall to which it goes is $\mathrm{R}/2$ inside the Earth through a narrow grove before coming to rest momentarily. The grove, contain an ideal spring of spring constant $\mathrm{K}$and natural length $\mathrm{R}$, the value of $\text{K}$is $\text{(R}$ is radius of Earth and $\text{M}$ mass of Earth)