The mass density of a spherical galaxy varies as $${K \over r}$$ over a large distance ‘r’ from its centre. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as :

Train A and train B are running on parallel tracks in the opposite directions with speeds of 36 km/hour and 72 km/hour, respectively. A person is walking in train A in the direction opposite to its motion with a speed of 1.8 km/ hour. Speed (in ms^–1) of this person as observed from train B will be close to :
(take the distance between the tracks as negligible)

If speed V, area A and force F are chosen as fundamental units, then the dimension of Young’s modulus will be

A small block starts slipping down from a point B on an inclined plane AB, which is making an angle $$\theta $$ with the horizontal section BC is smooth and the remaining section CA is rough with a coefficient of friction $$\mu $$. It is found that the block comes to rest as it reaches the bottom (point A) of the inclined plane. If BC = 2AC, the coefficient of friction is given by $$\mu $$ = ktan $$\theta $$ . The value of k is _________.