Quality factor of driven damped oscillating pendulum

[woodenbox]

Homework Statement

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?

Homework 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

The Attempt at a Solution

ω0 = T/(2*pi) = 1/(2*pi)

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.

 

[anathema]

Thank you for your post. Let's start by breaking down the problem into smaller parts.

Firstly, let's calculate the value of ω0, the natural frequency of the clock's pendulum. As you correctly stated, this can be done using the equation ω0 = sqrt(k/m), where k is the spring constant and m is the mass of the pendulum. In this case, the spring constant can be found using Hooke's Law, F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from equilibrium. In this case, the restoring force is equal to the weight of the pendulum, mg, and the displacement from equilibrium is the amplitude of the swing, 0.2 rad. Therefore, we can write:

mg = -k(0.2 rad)

Solving for k, we get k = -5mg. Substituting this into our equation for ω0, we get:

ω0 = sqrt((-5mg)/m) = sqrt(-5g)

Next, let's calculate γ, the damping coefficient. This can be done using the equation γ = b/m, where b is the damping constant and m is the mass of the pendulum. In this case, the damping constant can be found using the equation for the resonance width, γ, which is equal to the difference between the natural frequency, ω0, and the driving frequency, ω. We know that the driving frequency is equal to ω0 when the clock is at resonance, so we can write:

γ = ω0 - ω0 = 0

Therefore, we can conclude that the damping coefficient, γ, is equal to zero. This means that the clock is undamped, and there is no energy dissipation due to friction.

Finally, let's calculate the quality factor, Q. As you mentioned, Q can be calculated using the equation Q = ω0/γ. In this case, since γ is equal to zero, we can conclude that Q is equal to infinity. This means that the clock's pendulum will continue to swing with the same amplitude for an infinite amount of time.

Now, let's move on to the second part of the problem, which asks how long the clock would run if it were powered by a battery with 1 J capacity. To solve this, let's first calculate the total energy stored