The planets with radii R₁ and R₂ have densities ρ1, ρ2 respectively. Their atmospheric pressure is p1 and p2 respectively. Therefore, the ratio of masses of their atmospheres, neglecting variation of g within the limits of the atmosphere is

$\frac{ρ_{1} R_{2} ρ_{1}}{ρ_{2} R_{1} ρ_{2}}$

$\frac{ρ_{1} R_{2} ρ_{2}}{{pP}_{2} R_{1} ρ_{1}}$

$\frac{p_{1} R_{1} ρ_{1}}{p_{2} R_{2} ρ_{2}}$

$\frac{p_{1} R_{1} ρ_{2}}{p_{2} R_{2} ρ_{1}}$

Solution

$\frac{p_{1} R_{1} ρ_{2}}{p_{2} R_{2} ρ_{1}}$

Acceleration due to gravity

$g = \frac{GM}{R^{2}} = \frac{4}{3} πρGR ∴ \frac{g_{1}}{g_{2}} = \frac{ρ_{1} R_{1}}{ρ_{2} R_{2}}$

Atmospheric pressure can be given by

P = W/S

Where w = weight of the atmosphere,

S = surface area of the planet

$\frac{m_{1}}{m_{2}} = \frac{ρ_{1} S_{1} g_{2}}{ρ_{2} S_{2} g_{1}} = \frac{p_{1} . (4 {πR}_{1}^{2})}{p_{2} (4 {πR}_{2}^{2})} . \frac{ρ_{2} R_{1}}{ρ_{1} R_{1}} = \frac{p_{1} R_{1} ρ_{2}}{p_{2} R_{2} ρ_{1}}$

Some More Questions From Gravitation Chapter

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?

Which of the following statements is correct?
(a) Centre of the mass of a body always coincides with the centre of gravity of the body.
(b) Centre of the mass of a body is the point at which the total gravitational torque on the body is zero
(c) A couple on a body produce both translational and rotational motion in a body.
(d) Mechanical advantage greater than one means that small effort can be used to lift a large load.

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Gravitation

Some More Questions From Gravitation Chapter

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?

Two astronauts are floating in gravitational free space after having lost contact with their spaceship. The two will:

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