The ratio of energy required to raise a satellite of mass '$$m$$' to height '$$h$$' above the earth's surface to that required to put it into the orbit at same height is [ $$\mathrm{R}=$$ radius of earth]

A pendulum is oscillating with frequency '$$n$$' on the surface of the earth. It is taken to a depth $$\frac{R}{2}$$ below the surface of earth. New frequency of oscillation at depth $$\frac{R}{2}$$ is

[ $$R$$ is the radius of earth]

When the value of acceleration due to gravity '$$g$$' becomes $$\frac{g}{3}$$ above surface of height '$$h$$' then relation between '$$h$$' and '$$R$$' is ( $$\mathrm{R}=$$ radius of earth)

A particle of mass '$$m$$' is kept at rest at a height $$3 R$$ from the surface of earth, where '$$R$$' is radius of earth and '$$M$$' is the mass of earth. The minimum speed with which it should be projected, so that it does not return back is ( $$g=$$ acceleration due to gravity on the earth's surface)