Six stars of equal masses $M$ are moving about the centre of mass of the system such that they are always on the vertices of a regular hexagon of side length a. If their time period of motion is $2\pi \sqrt{\frac{N\sqrt{3}{a}^3}{GM\left(5\sqrt{3}+N\right)}}$ then value of $N$ is ____.

A planet is revolving around the sun in an elliptical orbit. If the speed of planet at shown position $P$ ($M$:Mass of Sun) is $\sqrt{\frac{kGM}{2\mathrm{a}}}$, then value of $k$ is ______.

A ring of mass $M$ and radius $R$ is fixed and a point mass $m$ is released from rest as shown in figure. If speed of the particle when it reaches the centre of the ring is $\sqrt{\left(k-\frac{k}{\sqrt{3}}\right)\frac{GM}{R}}$, then value of $k$ is ______.

A uniform solid sphere of mass $M$ is placed touching the uniform rod of mass $M$ and length $R$. If the rod rotates about the axis $AB$ with angular speed ${\omega }_0$ under the action of gravitational force of sphere, then