the total energy is the sum of the linear and rotational energy:
K = K_rot + K_lin
first, we find the rotational kinetic energy of a rotating disc with an angular velocity of w. see the references for the moment of inertia of a disc.
K_rot = (1/2)(I)(w^2)
I = (1/2)(m)(r^2)
K_rot = (1/4)(m)(r^2)(w^2)
next, we find the linear kinetic energy of a rolling disc:
K_lin = (1/2)(m)(v^2)
v = angular velocity circumference
= w (pi 2 r)
K_lin = (1/2)(m)(w2pir)^2
= (2pi^2)(m)(r^2)(w^2)
we find the total kinetic energy:
K = K_rot + K_lin
= (1/4)(m)(r^2)(w^2) + (2pi^2)(m)(r^2)(w^2)
and find the rotational contribution:
K_rot = K [K_rot/K]
K_rot = K [K_rot/(K_rot+K_lin)]
K_rot = K (1/4) / [(1/4) + (2pi^2)]
= K / (8pi^2 + 1)