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Kinetic Energy of Rotating Solid disk or cylinder about symmetry axis

  1. Apr 7, 2014 #1
    1. The problem statement, all variables and given/known data
    A horizontal 845.0 N merry-go-round with
    a radius of 4.3 m is started from rest by a
    constant horizontal force of 74.0 N applied
    tangentially to the merry-go-round. Find the
    kinetic energy of the merry-go-round after
    2.2 s. Assume it is a solid cylinder. The
    acceleration of gravity is 9.81 m/s2

    2. Relevant equations
    I of MGR = 1/2mR²
    angular KE = 1/2Iw2
    F = ma
    linear KE = 1/2mv²
    τ=αI = FR

    3. The attempt at a solution
    weight of MGR = 845 N
    mass of MGR = 845/9.81 = 86.1366 kg
    R = 4.3 m
    I of MGR = 1/2mR² = (0.5)(86.1366 kg)(4.3 m)² = 796.3328 kgm²
    applied tangential force (F) = 74 N
    time F is applied = t = 2.2 s

    Torque applied = (74 N)(4.3 m) = Iα
    α = 318.2 N·m/796.3328 kgm² = 0.39958 rad/s²
    w = αt = (0.39958 rad/s²)(2.2 s) = 0.8790797 rad/s
    angular KE = 1/2Iw² = (0.5)(796.3328 kgm²)(0.8790797 rad/s)² = 307.695468 J

    F = ma
    a = F/m = 74 N/86.1366 kg = 0.85910054 m/s²
    v = at = (.85910054 m/s² )(2.2 s) = 1.8900212 m/s
    linear KE = 1/2mv² = (0.5)(86.1366 kg)(1.8900212 m/s)² = 153.8477 J

    TOTAL KE = 1/2Iw² + 1/2mv² = 307.695468 J + 153.8477 J = 461.5432 J

    Possible reason for missing problem: Not sure if this is correct, but I think that there is no linear KE because the center of mass of the object is not moving. Therefore, I think the correct answer is 307.695468 J. I don't know if this line of reasoning is correct or not?
     
  2. jcsd
  3. Apr 7, 2014 #2

    maajdl

    User Avatar
    Gold Member

    Of course, linear KE = 0 .

    The axis of the MGR is fixed.
    If the axis would not be fixed, you "angular+linear" calculation would be wrong anyway.
    You would then need to specify the problem more precisely: what would be the meaning of a tangential force?
    How could a tangential force accelerate the center of mass and put the MGR into rotation both together at the rate would indicated?
     
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