Physics - Center of Mass Question with Solution | TestHub

The square plate has mass $M_{plate}=2M$ and its center of mass is at $(0,0)$ as given.

The rod has mass $M_{rod}=M$ and length $L$. It is attached perpendicularly to the center of one side of the square plate.

If the square plate is centered at the origin, its sides extend from $-L/2$ to $L/2$. Let the rod be attached to the side $x=L/2$ and extend along the positive X-axis. The center of mass of the rod itself is at a distance $L/2$ from its attachment point. So, the COM of the rod is at $(L/2+L/2,0)=(L,0)$. The x-coordinate of the center of mass of the combined system is $X_{CM}=\frac{M_{plate}{x}_{plate}+M_{rod}{x}_{rod}}{M_{plate}+M_{rod}}=\frac{(2M)(0)+M(L)}{2M+M}=\frac{ML}{3M}=L/3$. The y-coordinate of the center of mass of the combined system is $Y_{CM}=\frac{M_{plate}{y}_{plate}+M_{rod}{y}_{rod}}{M_{plate}+M_{rod}}=\frac{(2M)(0)+M(0)}{2M+M}=0$. The center of mass of the combined system is $(L/3,0)$.