For a system consisting of two objects,when can they be treated as point particles?

$A$ large number of particles are placed around the origin, each at a distance $R$ from the origin. The distance of the center of mass of the system from the origin is

The integral $\int \vec{r} dm = 0$ for a homogeneous body is defined for which point?

The sum of moments of all the particles in a system about its center of mass is always . . . . . .

The velocities of three particles of masses $20 \ g$,$30 \ g$ and $50 \ g$ are $10 \hat{i}$,$10 \hat{j}$ and $10 \hat{k}$,respectively. The velocity of the centre of mass of the three particles is

The center of mass of a system of particles with masses $1 \, g, 2 \, g$,and $3 \, g$ is at the origin. When a particle of mass $4 \, g$ with position vector $\alpha(\hat{i} + 2\hat{j} + 3\hat{k})$ is added,the center of mass of the system becomes $(1, 2, 3)$. If $\alpha$ is a constant,its value must be: