$A$ spherical shell of $1 \, kg$ mass and radius $R$ is rolling with angular speed $\omega$ on a horizontal plane (as shown in the figure). The magnitude of the angular momentum of the shell about the origin $O$ is $\frac{a}{3} R^{2} \omega$. The value of $a$ will be ..............

The speed of rolling of a ring of mass $M$ changes from $V$ to $3\ V$. What is the change in its kinetic energy (in $,MV^2$)?

$A$ small roller of diameter $20 \ cm$ has an axle of diameter $10 \ cm$. It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle. The scale is now pushed slowly on the axle so that it moves without slipping on the axle,and the roller starts rolling without slipping. After the roller has moved $50 \ cm$,the position of the scale will look like (figures are schematic and not drawn to scale)-

$A$ circular disc of mass $0.41 \ kg$ and radius $10 \ m$ rolls without slipping with a velocity of $2 \ m/s$. The total kinetic energy of the disc is ....... $J$.

Two identical circular loops are moving with the same kinetic energy; one rolls and the other slides. The ratio of their speed is:

The ratio of kinetic energies of two spheres rolling with equal centre of mass velocities is $2 : 1$. If their radii are in the ratio $2 : 1$,then the ratio of their masses will be