rotation

if the fraction of total energy associated with pure rolling is A for a ring and B for a solid sphere....find A & B

For a ring I will give you the solution, you try for the solid sphere.

For a ring

1/2Iω² = R (rotational kinetic energy)
1/2mv² = T (translational kinetic energy)

also I=mr²

also I=mr²
R = 1/2mr²Ï‰²
R = 1/2m(rω)²
because it is pure rolling rω = v

thus R= 1/2 mv² = T if you see closely

total energy - 2T

thus the ratio is 1/2

Now will you try for the other part?