The density of a solid sphere of mass M and radius R varies with distance r from its centre as ρ=kr where k is a constant. The gravitational field due to this sphere at a distance 2R from its centre is given by πkGN . Then the value of N is

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The density of a solid sphere of mass M and radius R varies with distance r from its centre as $ρ = \frac{k}{r}$ where k is a constant. The gravitational field due to this sphere at a distance 2R from its centre is given by $\frac{π k G}{N}$ . Then the value of $N$ is

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$dm = ρ (4 π r^{2} d r) = 4 π k rdr$

$m = \int_{0}^{R} dm = \int_{0}^{R} 4 π k rdr = 2 π k R^{2}$

$Thus, gravitational field at a distance of 2 R from the center is$

$g = \frac{Gm}{(2 R)^{2}} = \frac{π Gk}{2}$

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