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4 March, 19:37

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

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  1. 4 March, 20:05
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    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)
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