GATE-2018 (24) :Wyllie – Time average equation

22. In order to have the same value of the surface acceleration due to gravity as on earth, a planet twice its radius must have a mean relative density of.
 

1)2.75                 2)5.5                      3)1.4                      4)2.25

 

Formula:

acceleration due to gravity g= (G*m) /r² -----------------(1)

 

Form the above problem, we need a relation between the radius and density

 

So,

 

Density = mass / Volume

 

mass= Density * Volume

 

m= ρ * v ---------------(2)

 

Substitute eq (2) in eq (1)

 

g = (G*ρ*v) / r²

 

then,

 

(G*ρ₁*v₁) / r₁² = (G*ρ₂*v₂) / r₂² -------------------(3)

 

Volume of the Earth and planets are v= 4/3 * (П* r³)

 

substitute in equation (3)

 

(G*ρ₁* 4/3 * (П* r₁³) ) / r₁² = (G*ρ₂* 4/3 * (П* r₂³) ) / r₂²

 

Therefore, simplifying above equation, we get,

 

ρ₁*r₁ = ρ₂* r₂

 

(ρ₂ /ρ₁) = (r₁ /r₂) ----------------(4)

 

from above problem

 

r₂= 2*r₁

 

ρ₁= 5.5 gm/cc earth density

 

substitute above values in equation (4)

 

(ρ₂ /ρ₁) = (r₁ /2*r₁)

            = 1/2

 

ρ₂= 5.5 /2

 

ρ₂ =2.75 gm/cc