Three bodies,a ring,a solid disc,and a solid sphere,roll down the same inclined plane without slipping. The radii of the bodies are identical,and they start from rest. If $V_S, V_R$,and $V_D$ are the speeds of the sphere,ring,and disc,respectively,when they reach the bottom,then the correct option is:

If a sphere is rolling,the ratio of its rotational energy to the total kinetic energy is given by

Two uniform thin spherical shells are made from different materials. Both shells have a mass of $2 \,kg$ and an outer radius of $20 \,cm$. When they are both rolled down the same inclined plane without slipping,the times they take to cover equal distances differ by $1 \%$. If the thickness of the thinner shell is $0.5 \,cm$,the thickness of the other one is closest to ........... $\,cm$.

$A$ small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches up to a maximum height of $3v^2/4g$ with respect to the initial position. The object is

The following bodies,
$(1)$ a ring
$(2)$ a disc
$(3)$ a solid cylinder
$(4)$ a solid sphere,
of same mass $m$ and radius $R$ are allowed to roll down without slipping simultaneously from the top of an inclined plane. The body which will reach first at the bottom of the inclined plane is ...........
[Mark the body as per their respective numbering given in the question]

Suppose a body of mass $M$ and radius $R$ is allowed to roll on an inclined plane without slipping from its topmost point $A$ at a height $h$. The velocity acquired by the body,as it reaches the bottom of the inclined plane,is given by $\beta = 1 + \frac{I}{MR^2}$. Find the expression for the velocity $v$.