Two particles of masses $m_1$ and $m_2$ are separated by a distance $d$. The shift in the centre of mass when the two particles are interchanged is:

The figure shows two cylindrical rods whose centers of mass are marked as $A$ and $B$. The line $AB$ divides the region into two parts: one containing point $O$ (region $1$) and the other containing point $O'$ (region $2$). Choose the correct option regarding the center of mass of the combined system.

Three particles of masses $20 \, g$,$30 \, g$,and $50 \, g$ have velocities $10 \, \hat{i}$,$10 \, \hat{j}$,and $10 \, \hat{k}$ respectively. The velocity of the center of mass of these three particles is:

$A$ uniform thin rod of length $L$ and mass $M$ is placed along the $x$-axis with one end at the origin. Find the position of the centre of mass of the rod.

Consider the following statements regarding the Centre of Mass $(CM)$ of objects of radius $R$ from their geometric centre:
$[1]$ $CM$ of a uniform semicircular disc is at $2R/\pi$.
$[2]$ $CM$ of a uniform semicircular ring is at $4R/3\pi$.
$[3]$ $CM$ of a solid hemisphere is at $4R/3\pi$.
$[4]$ $CM$ of a hemispherical shell is at $R/2$.
Which of these statements are correct?

Two particles whose masses are $10\,kg$ and $30\,kg$ and their position vectors are $\hat{i} + \hat{j} + \hat{k}$ and $-\hat{i} - \hat{j} - \hat{k}$ respectively would have the centre of mass at: