Two bodies,a ring and a solid cylinder of the same material,are rolling down without slipping an inclined plane. The radii of the bodies are the same. The ratio of the velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is $\frac{\sqrt{x}}{2}$. Then,the value of $x$ is .... .

$A$ ball rolls down an inclined plane,as shown in the figure. The ball is first released from rest from $P$ and then later from $Q$. Which of the following statement$(s)$ is/are correct?
$(i)$ The ball takes twice as much time to roll from $Q$ to $O$ as it does to roll from $P$ to $O$.
$(ii)$ The acceleration of the ball at $Q$ is twice as large as the acceleration at $P$.
$(iii)$ The ball has twice as much $K.E.$ at $O$ when rolling from $Q$ as it does when rolling from $P$.

$A$ solid sphere and a hollow cylinder roll up without slipping on the same inclined plane with the same initial speed $v$. The sphere and the cylinder reach maximum heights $h_1$ and $h_2$,respectively,above the initial level. The ratio $h_1: h_2$ is $\frac{n}{10}$. The value of $n$ is . . . . . . .

$A$ disc of mass $m$ and radius $r$ rolls down an inclined plane of height $h$. When it reaches the bottom of the plane,its rotational kinetic energy is ($g=$ acceleration due to gravity).

$A$ solid sphere is rolling without slipping on a semi-circular track of radius $R = 10 \ m$ as shown in the figure. The radius of the solid sphere is much smaller than the radius of the semi-circular track. At the lowest point,it has a velocity $v = 10 \ m/s$. To what maximum angle $\theta$ from the vertical will the sphere travel before it comes back down? Neglect the rolling friction between the sphere and the track. (Take $g = 10 \ m/s^2$)

$A$ uniform solid spherical ball is rolling down a smooth inclined plane from a height $h$. The velocity attained by the ball when it reaches the bottom of the inclined plane is $v$. If the ball is now thrown vertically upwards with the same velocity $v$,the maximum height to which the ball will rise is