# Grandfather's clock

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?

• In which you chapter you found the problem? I Have the same book. I have nearly completed it but didn’t find the problem which you stated above Jan 31, 2022 at 17:04
• problem 11.10 in the 6th edition Feb 1, 2022 at 14:02

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 $$\theta$$.
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.