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

Consider the arrangement shown in figure. A thin rod (mass $M$ and length $4R$ ) and a sphere of equal mass $M$ and radius $R$ are located as shown in figure. The centre of the sphere being at a distance $2R$ from the centre of the rod on its perpendicular bisector. Assume that the system of rod and sphere is in gravity free space. Find the force acting on the rod due to sphere.