 Solution

Let $X$ increase along the handle, with zero at the centre of the circular head. The ring has its centre of mass at $X=0$. The rod occupies $a\le X\le a+l$, so its centre of mass is at $a+l/2$. The mass-weighted mean is thereforeThis uses the source diagram's rim-to-end handle length $l$, not a rod beginning at the head's centre.