$A$ study of binary stars is most helpful in

$A$ particle of mass $M$ is situated at the centre of a spherical shell of same mass and radius $a.$ The magnitude of the gravitational potential at a point situated at $a/2$ distance from the centre,will be

Two bodies,each of mass $M$,are kept fixed with a separation $2L$. $A$ particle of mass $m$ is projected from the mid-point of the line joining their centres,perpendicular to the line. The gravitational constant is $G$. The correct statement$(s)$ is/are:
$(a)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $4 \sqrt{\frac{GM}{L}}$
$(b)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $2 \sqrt{\frac{GM}{L}}$
$(c)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $\sqrt{\frac{2GM}{L}}$
$(d)$ The energy of the mass $m$ remains constant

As shown in the figure,two spherical cavities are made in a uniform solid sphere of radius $R$. The boundaries of the cavities touch at the centre of the sphere. The centers of the cavities and the sphere lie on the $X$-axis. The mass of the solid sphere before the cavities were created was $M$. The gravitational force on a point mass $m$ at a distance $d$ away from the centre of the solid sphere is

Three identical spheres of mass $m$ are placed at the vertices of an equilateral triangle of side length $a$. When released,they interact only through gravitational force and collide after a time $T = 4 \text{ s}$. If the sides of the triangle are increased to length $2a$ and the masses of the spheres are made $2m$,then they will collide after . . . . . . seconds.

Three point masses,each of mass $m$,are kept at the corners of an equilateral triangle of side $L$. The system rotates about the center of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to