Asolid 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.

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)

the one with the highest speed

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