Define the position vector of centre of mass.

A circular disc of radius $R$ is removed from a bigger uniform circular disc of radius $2R$ such that the circumferences of the discs coincide. The centre of mass of the new  disc is $\alpha R$ from the centre of the bigger disc. The value of $\alpha$ is

A uniform circular disc of radius $a$ is taken. A circular portion of radius $b$ has been removed from it as shown in the figure. If the centre of hole is at a distance $c$ from the centre of the disc, the distance $x_2$ of the centre of mass of the remaining part from the initial centre of mass $O$ is given by

Figure shows two cylindrical rods whose center of mass is marked as $A$ and $B$ . Line $AB$ divides the region in two parts one containing point $O$ (region $1$ ) and other containing point $O'$ (region $2$ ). Choose the correct option regarding the center of mass of the combined system.

The position vector of three particles of masses $1\, kg, 2\, kg$ and $3\, kg$ are $\overrightarrow {{r_1}}  = (\widehat i + 4\widehat j + \widehat k)\,m,\overrightarrow {{r_2}}  = (\widehat i + \widehat j + \widehat k)\,m$ and $\overrightarrow {{r_3}}  = (2\widehat i - \widehat j - 2\widehat k)\,m$  respectively. The position  vector of their centre of mass is

Find the centre of mass of a uniform L-shaped lamina (a thin flat plate) with dimensions as shown. The mass of the lamina is $3 \;kg$.