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.

$Assertion$ : The Earth is slowing down and as a result the Moon is coming nearer to it.
$Reason$ : The angular momentum of the Earth-Moon system is not conserved.

$A$ man of mass $m$ starts falling towards a planet of mass $M$ and radius $R.$ As he reaches near to the surface,he realizes that he will pass through a small hole in the planet. As he enters the hole,he sees that the planet is really made of two pieces: a spherical shell of negligible thickness of mass $\frac{2M}{3}$ and a point mass $\frac{M}{3}$ at the centre. The change in the force of gravity experienced by the man is:

Particles of masses $m, 2m, 3m$ and $4m$ are placed at the corners of a square of side $a$. What is the net gravitational force on a particle of mass $m$ placed at the center?

Two identical thin uniform rods of mass $M$ and length $L$ are placed in a line at a separation of $L$. Find the gravitational force acting between them.

Three identical particles $A, B$,and $C$ of mass $100 \, kg$ each are placed in a straight line with $AB = BC = 13 \, m$. The gravitational force on a fourth particle $P$ of the same mass is $F$,when placed at a distance $13 \, m$ from the particle $B$ on the perpendicular bisector of the line $AC$. The value of $F$ will be approximately $....G$.