A uniform disc of radius $R$ is put over another uniform disc of radius $2R$ of the same thickness and density. The peripheries of the two discs touch each other. Locate the centre of mass of the system.

A square plate of edge $d$ and a circular disc of diameter $d$ are placed touching each other at the midpoint of an edge of the plate, as shown in the figure. Locate the centre of mass of the combination, assuming same mass per unit area for the two plates.

Calculate the velocity of the centre of mass of the system of particles shown in the figure.

Two blocks of masses $10 \mathrm{kg}$ and $30 \mathrm{kg}$ are placed along a vertical line. The first block is raised through a height of $7 \mathrm{cm}$. By what distance should the second mass be moved to raise the centre of mass by $1 \mathrm{cm}$.

Consider a gravity-free hall in which a tray of mass $M,$ carrying a cubical block of ice of mass $m$ and edge $L,$ is at rest in the middle, as shown in the figure. If the ice melts, by what distance does the centre of mass of "the tray plus the ice" system descend?

Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_1$ and $R_2$, as shown in figure.