Spring 2006 1 Mech

A grandfather clock has a pendulum length of $1.0$ m and a bob of mass $m = 0.5$ kg. A mass of $2$ kg falls $0.7$ m in seven days to keep the amplitude of the pendulum oscillations steady at $0.03$ rad.

(a) The quality factor $Q$ of a damped oscillator is defined as

(1)
\begin{align} Q = 2\pi \times \frac{\text{average energy}}{\text{energy lost per cycle}} \end{align}

What is the $Q$ of the system?

(b) With no energy input from the falling mass, the pendulum obeys the small angle equation of motion

(2)
\begin{align} \ddot{\theta} + 2\beta\dot{\theta}+\omega_0^2 \theta = 0 \end{align}

Find $\omega_0$ and $\beta$

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Part (a), numerical answer should be $Q \sim 300$. Solution forgot to multiply by $2\pi$.