Center of Mass

Consider a thin uniform circular disk of radius \( r \). A square of side length \( \tfrac{r}{2} \) is cut out from this disk such that one of the vertices of the square that is cut out lies at the center of the disk. The center of mass of the disc with the hole in it is a distance \( d \) from the center of mass of the original disc. This distance \( d \) can be expressed in the form \( \tfrac{\sqrt{2}}{a\pi + b} r \) where \( a \) and \( b \) are integers. What is the value of \( a + b \)?

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