The distance between two stars of masses 3M_S and 6M_S is 9R. Here R is the mean distance between the centers of the Earth and the Sun, and M_S is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period nT, where T is the period of Earth's revolution around the Sun. The value of n is __________.

A large spherical mass M is fixed at one position and two identical masses m are kept on a line passing through the centre of M (see figure). The point masses are connected by a rigid massless rod of length l and this assembly is free to move along the line connecting them.


All three masses interact only through their mutual gravitational interaction. When the point mass nearer to M is at a distance r = 3l from M the tension in the rod is zero for m = $$k\left( {{M \over {288}}} \right)$$. The value of k is

A bullet is fired vertically upwards with velocity v from the surface of a spherical planet. When it reaches its maximum height, its acceleration due to the planet’s gravity is $${\left( {{1 \over 4}} \right)^{th}}$$ of its value at the surface of the planet. If the escape velocity from the planet is $${v_{esc}} = v\sqrt N $$, then the value of N is (ignore energy loss due to atmosphere)

A binary star consists of two stars A (mass 2.2M_s) and B (mass 11M_s), where M_s is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is