Four identical spheres each of mass $m$ are placed at the corners of a square of side $2 \, m$. Taking the point of intersection of the diagonals as the origin,the coordinates of the centre of mass are

Obtain the general expression for the centre of mass of a system of $n$ distributed particles in three dimensions.

$(a)$ Centre of gravity $(C.G.)$ of a body is the point at which the weight of the body acts.
$(b)$ Centre of mass coincides with the centre of gravity if the earth is assumed to have an infinitely large radius.
$(c)$ To evaluate the gravitational field intensity due to any body at an external point,the entire mass of the body can be considered to be concentrated at its $C.G.$
$(d)$ The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the $C.G.$ of the body to the axis of rotation.
Which one of the following pairs of statements is correct?

Find the position of the center of mass of a system of two particles of masses $m_1$ and $m_2$ separated by a distance $L$.

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:

Choose the correct statement about the centre of mass $(CM)$ of a system of two particles.