A loop rolls down on an inclined plane. The fraction of its total kinetic energy that is associated with the rotational motion is

$A$ uniform rod $AB$ of length $L$ and mass $M$ is lying on a smooth table. $A$ small particle of mass $m$ strike the rod with a velocity $v_0$ at point $C$ a distance $x$ from the centre $O$. The particle comes to rest after collision. The value of $x$, so that point $A$ of the rod remains stationary just after collision, is :

A solid cylinder rolls without slipping down an inclined plane of height $h$. The velocity of the cylinder when it reaches the bottom is

Two uniform thin identical rods $AB$ and $CD$ each of mass $M$ and length $L$ are joined so as to form a cross as shown. The moment of inertia of the cross about a bisector line $EF$ is (Line $EF$ is perpendicular to $ABCD$ plane)

 A thin circular ring of mass $m$ and radius $R$ is rotating  bout its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $\omega '$  = 

Let $F $ be the force acting on a particle having position vector $\vec r$ and $\vec T$ be the torque of this force about the origin.  Then ..