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The density inside a solid sphere of radius $a$ is given by $ρ = ρ_{0} a / r$ , where $ρ_{0}$ is the density at the surface and $r$ denotes the distance from the centre. Find the gravitational field due to this sphere at a distance $2 a$ from its centre.

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$\frac{π G ρ_{0} a}{2}$

$\frac{π G ρ_{0} a}{3}$

$\frac{π G ρ_{0} a}{4}$

$\frac{π G ρ_{0} a}{6}$

answer is A.

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Detailed Solution

The density inside a solid sphere of radius a is given by rho=rho_(0)(1-r/a), where rho_(0) is the density at the surface and r denotes the distance from the centre. Find the gravitational

Volume of small strip $d V = (4 π r^{2}) d r$

Mass of this strip $d m = (ρ) (d V)$

$= (\frac{p_{0} e}{r}) (4 π r^{2}) d r = (4 π a p_{0}) (r d r)$

$∴$ Mass of whole sphere $= \int_{0}^{a} (4 π a ρ_{0}) r d r$

$m = (2 π a^{3} ρ_{0})$

$E = \frac{G m}{(2 a)^{2}} = \frac{G (2 π a^{3} p_{0})}{4 a^{2}}$

$= \frac{π G ρ_{0} a}{2}$

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