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Quality factor of driven damped oscillating pendulum

  1. Oct 31, 2011 #1
    1. The problem statement, all variables and given/known data

    A small cuckoo clock has a pendulum 25 cm long with a mass of 10 g and a period of 1 s. The clock is powered by a 200 g weight which falls 2 m between the daily windings. The amplitude of the swing is 0.2 rad. What is the Q (quality factor) of the clock? How long would the clock run if it were powered by a battery with 1 J capacity?

    2. Relevant equations

    F = ma --> d2θ/dt2 + γ*(dθ/dt) + (ω0^2)*θ = driving force***

    ω0 = sqrt(k/m)
    γ= b/m
    Quality factor Q = ω0/γ
    Also, Q = energy stored in oscillator/energy dissipated per radian

    3. The attempt at a solution

    ω0 = T/(2*pi) = 1/(2*pi)

    Using arclength s and angle θ, m(d2s/dt2) = -mgsinθ, and since s = L*θ where L is the length of the pendulum, m*L*d2θ/dt2 = -mgsinθ - b(dθ/dt) + driving force

    driving force = weight of falling mass = mg = .2*g
    work done by driving force = .2*g*2meters = .4*g

    the resonance width of the system = γ, and occurs when ω - ω0 = ±γ/2

    I think that using the initial conditions given, I should be able to solve for ω somehow and then, having already solved for ω0, use ω - ω0 = ±γ/2 to solve for γ, and then Q would just be ω0/γ. However, solving for ω would require solving the second order non-homogeneous differential equation starred (***) above, and this class isn't supposed to require knowledge of ODEs (goes up to Calc IV).

    The other option is to use energy and use the fact that in the steady state, the energy lost is all lost by the damping force, but this would again require having an equation of motion for x from which to get dx/dt, from which to find the energy lost by the damping force -bv, so I'm still at a loss of how to do this without actually solving for the equation of motion.

    Any help would be greatly appreciated!
    Thank you.
     
    Last edited: Oct 31, 2011
  2. jcsd
  3. Nov 1, 2011 #2

    rude man

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    Homework Helper
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    You don't need to solve any fancy equations to handle this problem.

    Start with the actual fundamental definition of Q. State this in words, not using any symbols or equations at all.
     
    Last edited: Nov 1, 2011
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