$A$ wheel is rolling straight on ground without slipping. If the axis of the wheel has speed $v$,the instantaneous velocity of a point $P$ on the rim,defined by angle $\theta$ relative to the vertical,relative to the ground will be:

$A$ symmetrical body of mass $M$ and radius $R$ is rolling without slipping on a horizontal surface with linear speed $v$. Then its angular speed is

$A$ solid sphere of mass $2 \ kg$ and radius $0.5 \ m$ is rolling without slipping on a horizontal surface. The ratio of the rotational and translational kinetic energies of the sphere is (in $: 5$)

$A$ uniform solid cylinder of mass $M = 3 \ kg$ and radius $R = 10 \ cm$ is connected about an axis through the centre of the cylinder to a horizontal spring with spring constant $k = 8 \ N/m$. The cylinder is pulled back,stretching the spring $x = 1 \ m$ from equilibrium. When released,the cylinder rolls without slipping. What is the speed of the center of the cylinder when it returns to equilibrium? .................. $m/s$

$A$ solid sphere and a solid cylinder,each of mass $M$ and radius $R$,are rolling with a linear speed $v$ on a flat surface without slipping. Let $L_1$ be the magnitude of the angular momentum of the sphere with respect to a fixed point $O$ on the surface along the path of the sphere. Likewise,let $L_2$ be the magnitude of the angular momentum of the cylinder with respect to the same fixed point $O$ along its path. The ratio $\frac{L_1}{L_2}$ is

$A$ uniform disc of mass $0.5\,kg$ and radius $r$ is projected with velocity $18\,m/s$ at $t = 0\,s$ on a rough horizontal surface. It starts off with a purely sliding motion at $t = 0\,s$. After $2\,s$ it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after $2\,s$ will be $..............J$ (given,coefficient of friction is $0.3$ and $g = 10\,m/s^2$).