Two point masses $m$ and $M$ are separated by a distance $L$. The distance of the centre of mass of the system from $m$ is

Consider a system of two particles of masses $m_1$ and $m_2$. If mass $m_1$ is pushed towards the center of mass of the system by a distance $d$,by what distance should the mass $m_2$ be displaced so that the center of mass of the system remains unchanged?

Three identical metal balls,each of radius $r$,are placed touching each other on a horizontal surface such that an equilateral triangle is formed when the centres of the three balls are joined. The centre of mass of the system is located at:

The linear mass density of a rod of length $L$ varies as $\lambda = A + Bx$. Find the center of mass.

$A$ 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:

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