The density of a solid spherical planet of radius R is given as ρ = ρ 0 r , where ρ 0 = constant and r is distance measured from centre of planet. The acceleration due to gravity of this planet is

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The density of a solid spherical planet of radius R is given as ρ = ρ 0 ​ r , where ρ 0 ​ = constant and r is distance measured from centre of planet. The acceleration due to gravity of this planet is half of maximum value at distance x from centre and also at a distance y from the centre. The value of x + y is R β α ​ ​ . Here α and β are single digit integer. Find the value of α − β ?

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d m ​ = ρ 0 ​ r 4 π r 2 d r ​ Inside g 1 ​ = g 1 ​ = x 2 G ρ 0 ​ π x 4 ​ g 1 ​ = G ρ 0 ​ π x 2 ​ Outside- g 2 ​ = y 2 Gm ​ = y 2 G ∫ 0 R ​ ρ 0 ​ × 4 π r 2 dr ​ = y 2 G ρ 0 ​ π R 4 ​ ​ Given g 1 ​ = g 2 ​ = 2 1 ​ g surface ​ g surface ​ = G ρ 0 ​ π R 2 G ρ 0 ​ π x 2 = y 2 G ρ 0 ​ π R 4 ​ = 2 1 ​ G ρ 0 ​ π R 2 ⇒ x = 2 ​ R ​ , y = 2 ​ R x + y = 2 ​ R ​ + 2 ​ R = 2 ​ 3 R ​ ​ ​ ⇒ x + y = R 2 9 ​ ​ α = 9 β = 2 ​

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