$A$ solid sphere of mass $500 \ g$ and radius $10 \ cm$ rolls without slipping with a velocity of $20 \ cm/s$. The total kinetic energy of the sphere will be ........ $J$.

$A$ solid cylinder having radius $R$ and length $L$ is slipping on a rough horizontal plane. At time $t = 0$ the cylinder has a translational velocity $v_0 = 49 \text{ m/s}$, perpendicular to its axis and a rotational velocity $v_0/4R$ about the centre. The time taken by the cylinder to start rolling is . . . . . . seconds. (coefficient of kinetic friction $\mu_K = 0.25$ and $g = 9.8 \text{ m/s}^2$)

$A$ uniform disc of radius $R$ is resting on a table on its rim. The coefficient of friction between the disc and the table is $\mu$. Now,the disc is pulled with a force $F$ applied at its center as shown in the figure. What is the maximum value of $F$ for which the disc rolls without slipping?

In a bicycle,the radius of the rear wheel is twice the radius of the front wheel. If $r_F$ and $r_r$ are the radii,and $v_F$ and $v_r$ are the speeds of the topmost points of the wheels respectively,then:

$A$ ball is thrown on a lawn in such a way that it initially slides with a speed $v_0$ without rolling. It gradually picks up rotational motion. The speed of the ball at which there will be rolling without slipping is

$A$ solid disc rolls clockwise without slipping over a horizontal path with a constant speed $v$. Then the magnitude of the velocities of points $A, B$ and $C$ (see figure) with respect to a standing observer are: