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A grandfather clock problem.

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data
    The pendulum of a grandfather's clock activates an escapement mechanism every time is passes through the vertical. The escapement is under tension (provided by a hanging weight) and gives the pendulum a small impulse distance l from the pivot. The energy transferred by the impulse compensates for the energy dissipated by friction, so that the pendulum swings with a constant amplitude.

    What is the impulse needed to sustain the motion of a pendulum of length L and mass m, with an amplitude of swing [tex]\theta_0[/tex] and quality factor Q?

    2. Relevant equations
    Q = (energy stored in oscillator)/(energy lost per radian)

    3. The attempt at a solution
    Suppose the oscillator starts with energy E and momentum
    [tex] p_{top} = \sqrt{2mE} [/tex]
    at the top of its swing. The energy lost per quarter period is then
    [tex] E \theta_0 / Q [/tex],
    thus the energy of the oscillator at the bottom of its first swing is
    [tex] E - E \theta_0 / Q = E(1 - \theta_0 / Q) [/tex],
    and the momentum at the bottom of the first swing is
    [tex] p_{bottom} = \sqrt{2 m E(1 - \theta_0 / Q)} [/tex].
    In order for the pendulum to maintain constant amplitude, the impulse I must satisfy:
    [tex] p_{bottom} + I = p_{top} \Rightarrow I = \sqrt{2mE} - \sqrt{2 m E(1 - \theta_0 / Q)} = \sqrt{2mE}(1 - \sqrt{1 - \theta_0 / Q}) [/tex].
    Now the energy of the oscillator is
    [tex] E = mgL(1-cos\theta_0) [/tex],
    and therefore the desired impulse is
    [tex] I = (1 - \sqrt{1 - \theta_0 /Q}) \sqrt{2m^2gL(1 - cos\theta_0)} [/tex].

    How's this look? I'm missing the factor l in my solution, which makes me think its wrong. What say you?
    Last edited: Nov 23, 2009
  2. jcsd
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