A space ship of mass $$2 \times 10^{4} \mathrm{~kg}$$ is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be (if $$g=10 \mathrm{~m} / \mathrm{s}^{2}$$ and radius of earth $$=6400 \mathrm{~km}$$ ):

If $$\mathrm{V}$$ is the gravitational potential due to sphere of uniform density on it's surface, then it's value at the center of sphere will be:-

The radii of two planets 'A' and 'B' are 'R' and '4R' and their densities are $$\rho$$ and $$\rho / 3$$ respectively. The ratio of acceleration due to gravity at their surfaces $$\left(g_{A}: g_{B}\right)$$ will be:

The time period of a satellite, revolving above earth's surface at a height equal to $$\mathrm{R}$$ will be

(Given $$g=\pi^{2} \mathrm{~m} / \mathrm{s}^{2}, \mathrm{R}=$$ radius of earth)