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(D) The volume of a spherical shell of thickness $dr$ and radius $r$ is $dV = 4\pi r^2 dr$.The mass of this shell is $dM = \rho dV = \left( 20 \frac{r

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$\pi$

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(D) The volume of a spherical shell of thickness $dr$ and radius $r$ is $dV = 4\pi r^2 dr$.The mass of this shell is $dM = \rho dV = \left( 20 \frac{r^2}{R^2} \right) \cdot 4\pi r^2 dr = \frac{80\pi r^4}{R^2} dr$.The total mass $M$ of the solid sphere is obtained by integrating from $0$ to $R$:$M = \int_0^R dM = \int_0^R \frac{80\pi r^4}{R^2} dr = \frac{80\pi}{R^2} \left[ \frac{r^5}{5} \right]_0^R = \frac{80\pi R^5}{5R^2} = 16\pi R^3$.For a point outside the sphere at a distance $r = 4R$,the sphere acts as a point mass $M$ at its centre. The gravitational field $E$ is given by:$E = \frac{GM}{r^2} = \frac{G(16\pi R^3)}{(4R)^2} = \frac{16\pi G R^3}{16R^2} = \pi GR$.Therefore,the ratio $\frac{E}{GR} = \pi$.

The density of a solid sphere of radius $R$ is $\rho(r) = 20 \frac{r^2}{R^2}$,where $r$ is the distance from its centre. If the gravitational field due to this sphere at a distance $4R$ from its centre is $E$ and $G$ is the gravitational constant,then the ratio of $\frac{E}{GR}$ is

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