$A$ $2\ kg$ body and a $3\ kg$ body are moving along the $x-$ axis. At a particular instant,the $2\ kg$ body has a velocity of $3\ m/s$ and the $3\ kg$ body has a velocity of $2\ m/s$. The velocity of the centre of mass at that instant is

$A$ small ball $B$ of mass $m$ is suspended with a light inelastic string of length $L$ from a block $A$ of the same mass $m$,which can move on a smooth horizontal surface as shown in the figure. The ball is displaced by an angle $\theta$ from the equilibrium position and then released. The displacement of the centre of mass of the $A+B$ system until the string becomes vertical is

Two particles of mass $m_1$ and $m_2$ are separated by a distance $d$. If the first particle is moved towards the centre of mass by a distance $x$,then the distance by which the second particle must be moved towards the centre of mass to keep the centre of mass at the same position is:

In a system,two particles of masses $m_1 = 3 \text{ kg}$ and $m_2 = 2 \text{ kg}$ are placed at a certain distance from each other. The particle of mass $m_1$ is moved towards the center of mass of the system through a distance of $2 \text{ cm}$. In order to keep the center of mass of the system at the original position,the particle of mass $m_2$ should move towards the center of mass by a distance of . . . . . . $\text{cm}$.

$A$ body $A$ of mass $M$ while falling vertically downwards under gravity breaks into two parts; a body $B$ of mass $\frac{2}{3} M$ and a body $C$ of mass $\frac{1}{3} M$. The centre of mass of bodies $B$ and $C$ taken together shifts compared to that of body $A$ towards

There are some passengers inside a stationary railway compartment. The track is frictionless. The centre of mass of the compartment itself (without the passengers) is $C_1$,while the centre of mass of the 'compartment plus passengers' system is $C_2$. If the passengers move about inside the compartment along the track,then: