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(D) Let the radius of the original sphere be $R = r$. The mass of the original sphere is $M = \rho \cdot \frac{4}{3} \pi r^3$.$A$ smaller sphere of di

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none of these

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(D) Let the radius of the original sphere be $R = r$. The mass of the original sphere is $M = \rho \cdot \frac{4}{3} \pi r^3$.$A$ smaller sphere of diameter $r$ is cut out,so its radius is $r' = \frac{r}{2}$.The mass of the removed sphere is $m = \rho \cdot \frac{4}{3} \pi (\frac{r}{2})^3 = \frac{M}{8}$.To maximize the distance of the center of mass of the remaining part from the original center,the smaller sphere must be cut such that its center is at the maximum possible distance from the original center,which is $d = R - r' = r - \frac{r}{2} = \frac{r}{2}$.The distance of the center of mass of the remaining part from the original center is given by $x = \frac{m \cdot d}{M - m}$.Substituting the values: $x = \frac{(\frac{M}{8}) \cdot (\frac{r}{2})}{M - \frac{M}{8}} = \frac{\frac{Mr}{16}}{\frac{7M}{8}} = \frac{Mr}{16} \cdot \frac{8}{7M} = \frac{r}{14}$.Since $\frac{r}{14}$ is not among the given options,the correct answer is 'none of these'.

$A$ sphere of diameter $r$ is cut from a sphere of radius $r$ such that the centre of mass of the remaining mass is at the maximum distance from the original centre. What is this distance?

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