The ratio of the total kinetic energy to the rotational kinetic energy for a rolling disc is:

$A$ solid sphere of radius $10 \ cm$ and mass $500 \ g$ is rolling without slipping with a velocity of $20 \ cm/s$. The total kinetic energy of the sphere is: (in $J$)

At time $t=0$,a disk of radius $1 \text{ m}$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha = \frac{2}{3} \text{ rad s}^{-2}$. $A$ small stone is stuck to the disk. At $t=0$,it is at the contact point of the disk and the plane. Later,at time $t=\sqrt{\pi} \text{ s}$,the stone detaches itself and flies off tangentially from the disk. The maximum height (in $\text{m}$) reached by the stone measured from the plane is $\frac{1}{2} + \frac{x}{10}$. The value of $x$ is. . . . . . . [Take $g=10 \text{ m s}^{-2}$.]

$A$ solid homogeneous sphere of mass $M$ and radius $r$ is moving on a rough horizontal surface,partly rolling and partly sliding. During this kind of motion of this sphere,

$A$ disc of radius $2\; m$ and mass $100\; kg$ rolls on a horizontal floor. Its centre of mass has a speed of $20\; cm/s$. How much work is needed to stop it?

Consider a cylinder of mass $M$ resting on a rough horizontal rug that is pulled out from under it with acceleration $a$ perpendicular to the axis of the cylinder. What is the friction force $F_{friction}$ at point $P$? It is assumed that the cylinder does not slip.