Centre of Mass

From a uniform disk of radius $R$, a circular hole of radius $R/2$ is cut out. The centre of the hole is at $R/2$ from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

A non-uniform bar of weight $W$ is suspended at rest by two strings of negligible weight as shown in the figure. The angles made by the strings with the vertical are $36.9°$ and $53.1°$ respectively. The bar is $2 \mathrm{m}$ long. Calculate the distance $d$ of the centre of gravity of the bar from its left end.

In the HCl molecule, the separation between the nuclei of the two atoms is about $1.27 \mathrm{\AA }$ $\left(1 \mathrm{\AA }={10}^{-10} \mathrm{m}\right)$. Find the approximate location of the CM of the molecule, given that a chlorine atom is about $35.5$ times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?