How Is the Impulse Calculated to Sustain a Grandfather Clock's Pendulum?

[fluxions]

Homework Statement

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 [tex]\theta_0[/tex] and quality factor Q?

Homework Equations

Q = (energy stored in oscillator)/(energy lost per radian)

The Attempt at a Solution

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

How's this look? I'm missing the factor l in my solution, which makes me think its wrong. What say you?

 

[jbsteele]

Your solution looks good, but you forgot to include the factor l. The impulse needed to sustain the motion of a pendulum of length L and mass m, with an amplitude of swing θ0 and quality factor Q is I = (1 - \sqrt{1 - \theta_0 /Q}) \sqrt{2m^2gL(1 - cos\theta_0)} + l .

 

[kerimek]

Your solution looks good overall, but you are correct in noticing that the factor l is missing. The impulse needed to sustain the motion of the pendulum also depends on the length of the pendulum, as it affects the amount of energy transferred by the escapement. So the final equation for the impulse should be:

I = l(1 - √(1 - θ0/Q)) √(2m^2gL(1 - cosθ0))

This accounts for the effect of the pendulum's length on the impulse needed to maintain constant amplitude. Great job on the solution!