2 planets have radii R1 and R2 and densities p1 and p2 respectively, what will be the ratio of acceleration due to gravity at their surface??
Answers
Answer:
Let the accelerations due to gravity on planets be g1 & g2
We know g = GM/R^2
Density of planets given as ρ1, ρ2
ρ = M/V = M/((4/3)πR^3)
M/R^2 = 4πρR/3
g = 4πGρR/3
g1 = 4πGρ1R1/3
g2 = 4πGρ2R2/3
Dividing g1 by g2, we get
g1 : g2 = ρ1R1 : ρ2R2
Given : two planets have radii R1 and R2 and their densities are ρ1 and ρ2 respectively
To Find : ratio of their acceleration due to gravities at their surface
Solution:
g = GM/r²
g₁ = GM₁/R₁²
Density = Mass/Volume
Volume = (4/3)πr³
=> ρ₁ = M₁/(4/3)πR₁³
=> M₁ = ρ₁ (4/3)πR₁³
g₁ = GM₁/R₁²
=> g₁ = Gρ₁ (4/3)πR₁³/R₁²
=> g₁ = Gρ₁ (4/3)πR₁
g₂= GM₂/R₂²
M₂ = ρ₂ (4/3)πr₂³
g₂ = Gρ₂ (4/3)πR₂
g₁ / g₂ = Gρ₁ (4/3)πR₁ / Gρ₂ (4/3)πR₂
=> g₁ / g₂ = ρ₁R₁ / ρ₂R₂
=> g₁ : g₂ = ρ₁R₁ : ρ₂R₂
ratio of their acceleration due to gravities is ρ₁R₁ : ρ₂R₂
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