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

Consider a ring where linear mass density $\lambda$ varies with $\theta$ as $\lambda ={\lambda }_0\left|\cos \theta \right|\left({\lambda }_0>0\right)$. The gravitational field at point $O$ will be