The linear mass density of a rod of length $L$ varies as $\lambda = kx^2$,where $k$ is a constant and $x$ is the distance from one end. The position of the centre of mass of the rod is:

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

If the linear density of a rod of length $3 \ m$ varies as $\lambda = 2 + x$,then the position of the centre of mass of the rod is

$A$ $T$-shaped object with dimensions shown in the figure is lying on a smooth floor. $A$ force $\vec{F}$ is applied at point $P$ parallel to $AB$,such that the object has only translational motion without rotation. Find the location of $P$ with respect to $C$.

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?

Four bodies of equal mass start moving with the same speed as shown in the figure. In which of the following combinations will the centre of mass remain at the origin?