Three solid spheres are made to move on a rough horizontal surface. Sphere $P$ is given a spin and released. Sphere $Q$ is given a forward linear velocity. Sphere $R$ is given linear and rotational motions as shown in the figure. Directions of the friction force on spheres $P, Q, R$ are respectively

$A$ 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? .................. $m/s$

$A$ wheel of radius $r$ rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road,a small piece of mud separates from the wheel at its highest point $B$ and drops at point $C$ on the road. The distance $AC$ will be

If a solid sphere is rolling without slipping on a horizontal plane,then the ratio of its rotational and total kinetic energies is

$A$ 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 ........ $J$.

$A$ boy is pushing a ring of mass $2 \ kg$ and radius $0.5 \ m$ with a stick as shown in the figure. The stick applies a force of $2 \ N$ on the ring and rolls it without slipping with an acceleration of $0.3 \ m/s^2$. The coefficient of friction between the ground and the ring is large enough that rolling always occurs,and the coefficient of friction between the stick and the ring is $(P/10)$. The value of $P$ is