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?
F = ma --> d2θ/dt2 + γ*(dθ/dt) + (ω0^2)*θ = driving force***
ω0 = sqrt(k/m)
Quality factor Q = ω0/γ
Also, Q = energy stored in oscillator/energy dissipated per radian
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!