Half of one percent
Everything on this page comes out of one relationship. The period of a pendulum goes as the square root of its length, so the rate goes as one over that square root, and a small fractional change in length gives about half that fractional change in rate with the sign flipped. One percent longer is half a percent slower.
Half a percent of a day is 432 seconds, which is the figure worth carrying around. Worked exactly rather than linearised it is 429 seconds a day slower for one percent longer and 435 seconds a day faster for one percent shorter — the asymmetry is the square root refusing to be a straight line, and it is why a big correction should be made in two goes rather than one.
What a thou is worth
On a 39.14 inch pendulum, moving the bob one thousandth of an inch changes the rate by 1.1 seconds a day. That is a useful number to hold in your head, because it converts a rate error straight into a distance. Six seconds a day is 5.4 thou of bob movement, which on a 32 thread rod is 0.17 of a turn — a nudge, not a spin.
It also tells you when to stop. If your seven day test can only resolve the rate to a third of a second a day, then the finest correction worth making is about three tenths of a thou, and there is no point turning a nut by five degrees.
Measuring, not guessing
The arithmetic here is exact and the measurement is not. Compare the clock against something better than another clock, at the same time of day, from a cold start at the same point in the winding cycle if you can manage it. Read to the second at both ends. Seven days divides your reading error by seven. Thirty days divides it by thirty, and by then you are usually measuring the room rather than the clock.
The temperature term
A steel rod grows when the hall warms up, and a longer pendulum runs slow. With 6.5 parts per million per degree Fahrenheit over a 20 degree swing, a 39.14 inch rod grows five thousandths of an inch and loses 5.6 seconds a day. Over a month of warm weather that is nearly three minutes, and it is entirely reversible when the weather turns.
This is why compensated pendulums exist and why a clock regulated in February is often out in August. The coefficient in the box is a field rather than a constant because it depends on what your rod is actually made of, and a wooden rod, an invar rod and a steel rod are three different clocks.
What this cannot see
Rate is not only length. Arc changes the period slightly and arc changes as the mainspring runs down or the case gets knocked. Depthing, pivot friction and a dirty movement all show up as rate that drifts rather than rate that is simply wrong. If a clock will not hold a rate no matter where the nut is, the problem is not the nut, and no amount of arithmetic on this page will find it.
Questions people ask
How long is a seconds pendulum?
About 39.1 inches, or 994 mm, as an ideal simple pendulum at 9.80665 m/s2. It beats once a second, meaning one tick per second and a full period of two seconds over and back. Real ones are set up a little longer than the formula because the mass is spread out along the bob and rod rather than concentrated at a point.
How many seconds a day does a one percent length change make?
About 432, because rate goes as one over the square root of length and half a percent of 86,400 seconds is 432. Worked exactly it is 429 seconds a day slower for one percent longer and 435 faster for one percent shorter. The two are not equal because the square root is not a straight line, which is also why a large correction is better made in two steps.
Which way do I turn the rating nut?
The page tells you which way the bob has to move — down to slow the clock, up to speed it up — and how far, in thousandths and in turns of the nut. It deliberately does not tell you which rotation does that, because the thread hand and the arrangement differ between movements. Mark the nut, turn it a little, see which way the bob went, and you know for that clock.
How far do I turn the nut for one second a day?
On a 39.14 inch pendulum with a 32 thread rod, one full turn is worth about 34.5 seconds a day, so one second a day is about 0.03 of a turn, roughly ten degrees. That is a smaller movement than most people expect and it is why regulating gets overdone. Work in fractions of a turn against a mark, not by feel.
Why does my clock keep time in winter and not in summer?
A steel rod expands as the room warms, the pendulum gets longer and the clock runs slow. At 6.5 parts per million per degree Fahrenheit, a 20 degree seasonal swing on a 39 inch rod is five thousandths of an inch and about 5.6 seconds a day, which is nearly three minutes over a month. Put your own rod coefficient in the box, because wood, steel and invar behave very differently.