A child is standing at one end of a long trolley moving with a speed  $v$  on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u,$  the centre of mass of the system (trolley + child) will move with a speed.

Two blocks which are connected to each other by means of a massless string are placed on two inclined planes as shown in fig. After releasing from rest, the magnitude of acceleration of the centre of mass of both the blocks is $(g = 10\, m/s^2)$

The moment of inertia of a sphere (mass $M$ and radius $R$) about it’s diameter is $I$. Four such spheres are arranged as shown in the figure. The moment of inertia of the system about axis $XX'$ will be

The moment of inertia of a thin circular lamina of mass $1\,kg$ and diameter $0.2\,metre$ rotating about one of its diameter is

The centre of mass of a body

Two particles of equal masses have velocities$\overrightarrow {{v_1}}  = 2\hat i\,m/s$ and $\overrightarrow {{v_2}}  = 2\hat j\,m/s$. The first particle has an acceleration $\overrightarrow {{a_1}}  = \left( {3i + 3\hat j} \right)\,m/{s^2}$,while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a