Centre of Mass of Continuous Bodies

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.

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?

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.

A disc of radius $R$ is cut out from a larger disc of radius $2R$ in such a way that the edge of the hole touches the edge of the disc. Locate the centre of mass of the residual disc.

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.

Seven homogeneous bricks, each of length $L,$ are arranged as shown in figure. Each brick is displaced with respect to the one in contact by $\frac{L}{10}$. Find the $x$-coordinate of the centre of mass relative to the origin shown.

In which of the following cases, the centre of the mass of a rod is certainly not at its centre.

A heavy ring of mass $m$ is clamped on the periphery of a light circular disc. A small particle having equal mass is clamped at the centre of the disc. The system is rotated in such a way that the centre moves in a circle of the radius $r$ with a uniform speed $v$. We conclude that an external force

A circular plate of diameter $d$ is kept in contact with a square plate of edge $d$ as shown in Figure. The density of the material and the thickness are same everywhere. The centre of mass of the composite system will be

A high-jumper successfully clears the bar. Is it possible that his centre of mass crossed the bar from below it? Try it with appropriate figures.

What can be said about the centre of mass of a uniform hemisphere without making any calculation? Will its distance from the centre be more than $\frac{R}{2}$ or less than $\frac{R}{2}$?

The weight $Mg$ of an extended body is generally shown in a diagram to act through the centre of mass. Does it mean that the earth does not attract other particles?