I am dealing with a grandfather's clock, taken from "Introduction To Mechanics" by D.Kleppner:
"The pendulum of a grandfather clock activates an escapement mechanism every time it passes through the vertical. The escapement is under tension (provided by a hanging weight) and gives the pendulum a small impulse a distance $l$ from the pivot. The energy transferred by this impulse compensates for the energy dissipated by friction, so that the pendulum swings with a constant amplitude.
(a) 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?
(b) Why is it desirable for the pendulum to engage the escapement as it passes vertical rather than at some other point of the cycle?"
I got (a) but I can't understand part (b). A naive thougt I had is that just like when pushing someone one a playground swing, the push would be most effective at the top hight. But the answer in the book for part (b) is the following:
"The point in the cycle where the impulse acts can vary due to mechanical imperfections. To minimize this effect, the impulse should be applied when v is not changing to first order with respect to $\theta$, which is at the bottom of the swing."
Why is the bold statement true?
Best Answer
The book answer confuses me a bit as well. First of all, it's not completely true - the direction of $v$ is constantly changing with respect to $\theta$, as it's always tangential to the arc of the pendulum. As $\theta$ changes, so must the direction of $v$. It is true, however, that the magnitude of $v$ isn't changing at the vertical, as this is the inflection point where the pendulum switches from gaining speed to losing speed. But there is no point where both the magnitude and direction $v$ are invariant with respect to $\theta$.
I thought this might be an Oberth Effect question, related to the notion that a fixed impulse can provide differing amounts of energy to objects moving at different speeds. A fixed impulse imparts a fixed momentum change, effectively a fixed velocity change when considering a fixed mass. But since kinetic energy is based on the square of the velocity, each additional increment of velocity adds more energy than the last - going from 1m/s to 2m/s requires the same impulse as going from 0m/s to 1m/s, but results in triple the kinetic energy change. If the escapement provides a fixed impulse regardless of the position of the pendulum, it will provide the maximum amount of energy if applied when the pendulum's velocity is highest, i.e., at the vertical. This will be the most "efficient" place to impart a fixed impulse, as it provides the most energy to the pendulum.
That the first-order derivative of the magnitude of the velocity with respect to $\theta$ is zero indicates that the pendulum velocity is either at a maximum or minimum at the vertical, but alone doesn't say which. I don't really understand the link to "mechanical imperfections" either - there is one theoretically most efficient point to apply impulse, regardless of how accurately you can hit that target.
I suppose there's an argument to be made against applying impulse at the extreme of the swing, as an early impulse could go against the pendulum's natural swing - this would be like an inelastic collision, where the the change in momentum results in an overall loss of system energy. One could make the argument that applying impulse at the vertical gives you the maximum amount of "error" in the impulse timing - you can be up to a quarter period early or late, and still have the impulse add KE rather than dissipate it. Applying impulse at the extreme, in contrast, might dissipate KE if applied early.
If this is an Oberth Effect question, the reasoning in the answer is incomplete as it is insufficient to show that the pendulum velocity is maximized at the vertical. The link to mechanical imperfections is also very weak, as you'd need to show the symmetry of the bob velocity about the vertical. The most efficient point for an impulse is the vertical, but being some amount early or late results in a symmetric loss in efficiency (applying impulse 0.1s early is the same as being 0.1s late, since velocities at those times are equal). Applying the impulse targeting any other point in the swing would impart different amounts of energy depending whether the timing was early or late (an impulse 0.1s early might impart more energy than an impulse 0.1s late, since velocities at those times are unequal).
Note that all of the reasoning in this answer supposes that the escapement provides a fixed impulse, but the problem statement never actually says that. If the escapement provides a fixed amount of energy, or applies a fixed force, the Oberth Effect doesn't apply.