The position vectors of two particles of masses $1 \ kg$ and $3 \ kg$ are $\hat{i} + 2\hat{j} + \hat{k}$ and $-3\hat{i} - 2\hat{j} + \hat{k}$ respectively. The position vector of the center of mass of this system is:

$A$ cricket bat is cut at the location of its centre of mass as shown in the figure. Then:

$A$ uniform thin rod $AB$ of length $L$ has linear mass density $\mu(x) = a + \frac{bx}{L}$,where $x$ is measured from $A$. If the $CM$ of the rod lies at a distance of $\frac{7}{12}L$ from $A$,then $a$ and $b$ are related as

$A$ uniform square wooden sheet of side $a$ has its centre of mass located at point $O$ as shown in the figure below on the left. $A$ square portion of side $b$ of this sheet is cut out to produce an $L$-shaped sheet as shown in the figure on the right. The centre of mass of the $L$-shaped sheet lies at the point $P$ (in the above diagram),when

$A$ uniform rod of length $L$ and mass $M$ is placed along the $x$-axis with its one end at the origin. The centre of mass of the rod is located at:

Where is the centre of mass of two isolated particles of equal mass located?