Question

Two planets are of the same material but their radii are in the ratio 2:1 then ratio of acceleration due to gravity on those two planets is

Answer 1

g1 = GM/(2R)^2
= GM/4R^2
g2= GM/R^2

Ratio = GM/4R^2 ÷ GM/R^2
= GM/4R^2 × R^2/GM
= 1/4
= 1:4


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Answer 2

Given : Two planets are of the same material but their radii are in the ratio 2:1

To find : Ratio of acceleration due to gravity on those two planets.

Solution :

Ratio of acceleration due to gravity on those two planets is 2:1

Let, the radius of smaller planet = R'

And, the radius of bigger planet = R"

As per the data mentioned in the question,

R"/R' = 2/1

(obtained from the ratio 2:1)

As, the planets are of same material so their density of material will be equal.

Let, the density of material of the both planets = ρ

Volume of smaller planet (spherical shape) = ⁴/₃ × π × (R')²

Volume of bigger planet (spherical shape) = ⁴/₃ × π × (R")³

Mass of smaller planet = Density of material of smaller planet × Volume of smaller planet = ρ × ⁴/₃ × π × (R')³

Mass of bigger planet = Density of material of bigger planet × Volume of bigger planet = ρ × ⁴/₃ × π × (R'')³

Acceleration due to gravity on bigger planet (g1) = G × Mass of the planet)/(Radius of the planet)² = [G × ρ × ⁴/₃ × π × (R'')³]/(R'')² = (G × ρ × ⁴/₃ × π × R'')

Acceleration due to gravity on smaller planet (g2) = (G × Mass of the planet)/(Radius of the planet)² = [G × ρ × ⁴/₃ × π × (R')³]/(R')² = (G × ρ × ⁴/₃ × π × R')

[In above mentioned calculations 'G' refers gravitational constant.]

The ratio of acceleration due to gravity on second planet :

= g1 : g2

= (G × ρ × ⁴/₃ × π × R'') ÷ (G × ρ × ⁴/₃ × π × R')

= R"/R'

= 2/1

= 2:1

Hence, the ratio of acceleration due to gravity on the two planets is 2:1