# A grandfather clock problem.

1. Nov 23, 2009

### fluxions

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

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