An ideal gas of molar mass M is contained in a very tall vertical cylindrical column in the uniform gravitational field. Assuming the gas temperature to be T, the height at which the centre of gravity of the gas is located is (R$$\to$$universal gas constant)

Assume that the earth moves around the sun in a circular orbit of radius R and there exists a planet which also move around the sun in a circular orbit with an angular speed twice as large as that of the earth. The earth of the orbit of the planet is

The ratio of accelerations due to gravity g₁ : g₂ on the surfaces of two planets is 5 : 2 and the ratio of their respective average densities $${{\rho _1}}$$ : $${{\rho _2}}$$ is 2 : 1. What is the ratio of respective escape velocities v₁ : v₂ from the surface of the planets?