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A weight of 50.0 N is suspended from a spring that has a force constant of 190 N/m. The system is undamped and is subjected to a harmonic driving force of frequency 10.0 Hz, resulting in a forced-motion amplitude of 4.00 cm. Determine the maximum value of the driving force.
N

  • physics -

    Force = m a
    w = 2 pi f = 62.8 radians/second
    Forcing function -kx = m d^2x/dt^2
    let forcing function = F sin wt
    so F sin w t = k x + m d^2x/dt^2
    let resulting motion be x = a sin wt + b cos w t
    then
    d^2x/dt^2 = - a w^2 sin wt - bw^2 cos wt
    so (s is sin and c is cos)
    Fsin wt=k(a s wt+b c wt)-m(a w^2 s wt-b w^2 c wt)

    sin terms
    F = k a - m a w^2
    F = a (k-w^2 m)
    but
    a = .04 meters
    k = 190 N/m
    m = 50/9.8 = 5.1 kg
    w^2 = 3944 rad^2/s^2
    so
    F = .04(190 - 3944*5.1)
    = -797N

    the cos terms give you the natural frequency motion at w = sqrt(k/m)
    note we are driving at about 60 rad/sec whereas the natural frequency is about 6 rad/sec so it is not going to move much for a large force.

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