A cockroach of mass $\frac {M}{2}$ is start moving, with velocity $V$ on the circumference of a disc of mass $'M'$ and $'R',$ what will be angular velocity of disc?

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

A particle originally at rest at the highest point of $a$ smooth vertical circle is slightly displaced. It will leave the circle at $a$ vertical distance $h$ below the highest point, such that

Consider a two particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass through a distance $d$, by what distance should the second particle be moved, so as to keep the centre of mass at the same position?

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down an inclined plane of length $L$ and height $h$. What is the speed of its centre of mass when the cylinder reaches its bottom

Four particles of masses $1\,kg, 2 \,kg, 3 \,kg$ and $4\, kg$ are placed at the four vertices  $A, B, C$ and $D$ of a square of side $1\, m$. The coordinates of centre of  mass of the particles are