Question

A sphere of diameter r is cut from a solid sphere of radius r such that the centre of mass of remaining part be at maximum distance from original centre, then this distance is :

Answer 1

See diagram.

Let x = distance between COM from the original centre of the sphere,

after the small sphere is cut off from it.

Let the density of the sphere be = d

Mass of big sphere ( full ) = $\frac{4}{3}\times \pi r^3\times d$

Mas of the small sphere cut from it = $\frac{4}{3}\times \pi (\frac{r}{2}{)}^3\times d$

Mas of the remaining object = $\frac{7}{6}\pi r^3\times d$

The position of the COM of the complete sphere = 0

$-x\times \frac{7}{6}\pi r^{3}d+\frac{r}{2}\times \frac{4}{3}\pi (\frac{r}{2}{)}^3\times d$ = 0

$x=\frac{r}{14}$