A solid disk has a total kinetic energy k. what is its rotational kinetic energy krot if it's rolling without slipping?

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) (w*2*pi*r) ^2

= (2*pi^2) (m) (r^2) (w^2)

we find the total kinetic energy:

K = K_rot + K_lin

= (1/4) (m) (r^2) (w^2) + (2*pi^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) + (2*pi^2) ]

= K / (8*pi^2 + 1)