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:

Three particles of masses $50\, g$,$100\, g$,and $150\, g$ are placed at the vertices of an equilateral triangle of side $1\, m$ (as shown in the figure). The $(x, y)$ coordinates of the centre of mass will be

$(n - 1)$ equal point masses each of mass $m$ are placed at the vertices of a regular $n$-polygon. The vacant vertex has a position vector $a$ with respect to the centre of the polygon. Find the position vector of the centre of mass.

The centre of mass of a system of two particles of masses $m_1$ and $m_2$ separated by a distance $d$ is:

Center of mass of a system of three particles of masses $1 \, kg, 2 \, kg, 3 \, kg$ is at the point $(1 \, m, 2 \, m, 3 \, m)$ and center of mass of another group of two particles of masses $2 \, kg$ and $3 \, kg$ is at point $(-1 \, m, 3 \, m, -2 \, m)$. Where should a $5 \, kg$ particle be placed so that the center of mass of the system of all these six particles shifts to the center of mass of the first system?

$A$ carpenter has constructed a toy as shown in the adjoining figure. If the density of the material of the sphere is $12$ times that of the cone,the position of the centre of mass of the toy is given by: