Quality factor of driven damped oscillating pendulum

• woodenbox
In summary, the conversation discusses a small cuckoo clock with a 25 cm pendulum, a 200 g weight, and a 0.2 rad amplitude. The goal is to find the clock's Q (quality factor) and determine how long it would run with a 1 J battery. The conversation delves into potential equations and methods for solving, but ultimately concludes that the actual fundamental definition of Q does not require solving any equations and can be stated in words.

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)

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.

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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.

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1. What is the quality factor (Q) of a driven damped oscillating pendulum?

The quality factor (Q) of a driven damped oscillating pendulum is a measure of the efficiency of energy transfer within the system. It is defined as the ratio of the energy stored in the oscillating system to the energy dissipated per cycle.

2. How is the quality factor (Q) of a driven damped oscillating pendulum calculated?

The quality factor (Q) of a driven damped oscillating pendulum can be calculated by dividing the resonant frequency of the pendulum by the bandwidth, where the bandwidth is the width of the frequency response curve at the half power point.

3. What is the significance of the quality factor (Q) in a driven damped oscillating pendulum?

The quality factor (Q) in a driven damped oscillating pendulum is an indication of the sharpness of the resonance peak. A higher Q value means a sharper peak and therefore a more efficient energy transfer. It also determines the rate at which the pendulum will decay and eventually come to rest.

4. How does damping affect the quality factor (Q) of a driven damped oscillating pendulum?

Damping in a driven damped oscillating pendulum reduces the Q value by increasing the rate of energy dissipation. This results in a broader resonance peak and a less efficient energy transfer. However, a small amount of damping can help reduce the amplitude of oscillations and prevent the pendulum from swinging too wildly.

5. Can the quality factor (Q) of a driven damped oscillating pendulum be changed?

Yes, the quality factor (Q) of a driven damped oscillating pendulum can be changed by altering the damping coefficient, the driving frequency, or the amplitude of the driving force. A higher damping coefficient will result in a lower Q value, while a higher driving frequency or amplitude will increase the Q value.