Question

A solid disk has a total kinetic energy k. what is its rotational kinetic energy krot if it's rolling without slipping? express your answer in terms of the variable k.

Answer 1

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