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A spherically symmetric non-rotating body of radius R has varying volume mass density given by $ρ = ρ_{0} r$ where $ρ_{0}$ is a positive constant and r is distance from centre. For this situation mark the correct statement(s) with respect to gravitational field intensity.

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it is definitely zero at centre

it may be non zero at centre

as one moves out (radially) from centre it first increases non-linearly acquires maximum value at surface and then decreases non-linearly

both (a) and (c) are correct

answer is D.

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Detailed Solution

$ρ = ρ_{0} r; dm = ρ .4 π^{2} dr, dm = ρ_{0} 4 π r^{3} dr$

From gauss law in gravitation

$\int \vec{E} \cdot \vec{ds} = - 4 π G [m_{e n}]$

$\int Edscos 180 = - 4 π G \int_{0}^{r} dm$

$E. 4 π r^{2} = 4 π G \int ρ_{0} 4 π r^{3} dr$

$E = \frac{4 π G ρ_{0} r^{2}}{4} = π G ρ_{0} r^{2}$

$E \propto r^{2}$ Non-linearly inside the sphere if r = 0 at the centre E = 0

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