Question

two planets have radii r1 and 2r1 and their densities are d1 and 4d1 respectively. the ratio of their acceleration due to gravities is
a) 1:8
b) 8:1
c) 4:1
d) 1:4​

Answer 1

Given : two planets have radii r1 and 2r1 and their densities are d1 and 4d1 respectively

To Find : ratio of their acceleration due to gravities

a) 1:8

b) 8:1

c) 4:1

d) 1:4​

Solution:

g = GM/r²

g₁ = GM₁/r₁²

Density = Mass/Volume

Volume = (4/3)πr³

=> d₁ = M₁/(4/3)πr₁³

=> M₁ = d₁ (4/3)πr₁³

g₁ = GM₁/r₁²

=> g₁ = Gd₁ (4/3)πr₁³/r₁²

=> g₁ = Gd₁ (4/3)πr₁

g₂= GM₂/r₂²

M₂ = d₂ (4/3)πr₂³

g₂ = Gd₂ (4/3)πr₂

r₂  = 2r₁

d₂  = 4d₁

=> g₂ = G 4d₁(4/3)π 2r₁

=> g₂ = 8  Gd₁ (4/3)πr₁

=>  g₂ = 8 g₁

=> g₁  / g₂  = 1/8

=>  g₁  :  g₂  = 1 : 8

ratio of their acceleration due to gravities is 1 : 8

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