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The Pendulum Equation and how it applies to clocks

The Pendulum Equation is:

Where:
T = period (time for one full swing) in seconds
= the constant pi, 3.14159, used in geometry involving circles and arcs
L = length of the pendulum arm in metres (or feet)
g = acceleration due to gravity: 9.81 m/s2 (or 32.2 ft/s2)

Some notes on a pendulum system:
1. The mass on the end has no effect on the period.
2. The length of the arm is what affects the period..
3. The angle of release has no real effect on the period.
4. A pendulum movement is a resonant system with a one resonant frequency: 1 Hz, or once a second movement through one point.

A pendulum is a falling object, and is subject to the forces of gravity. Do you remember that all objects in a vacuum fall at the same speed? Or that a marble and a bowling ball both fall at the same speed? That is why the mass (weight) of the pendulum does not matter for the period, the acceleration due to gravity is the same for all objects. The force of gravity (it's value in the equation above) changes a little based on your elevation and a few other things, but for the sake of clocks the gravity value for the equation above is just fine.

Notice that the period T depends on the length of the pendulum directly. As L gets larger, so will T. So to slow down the clock, increase L by moving the weight on the pendulum lower. Conversely, to speed up the clock move the weight higher up.
This isn't a linear relation, since there is a square root in the equation. That means that to accurately adjust a clock, you have to average in many cycles (swings) of the pendulum. The section below shows a method to use for setting clock accuracy.

A good Pendulum discussion can be found here: http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html
Want an intensely scientific discussion of pendulum's and accuracy?? Click here: http://www.ubr.com/ftp/pendulum.pdf

For an even better discussion on "The pendulum equation and how it applies to clocks", see the paper excerpt below from David LaBounty, used by permission.

Clock Beat Rates
By David J. LaBounty CMC, FBHI

With the advent of the new digital timing machines it has become easier to rate a clock using its beat rate. Beat rate is defined as the number of blows of the escape wheel teeth as they fall on the lock faces of the pallets in a given time period. This time period is usually measured in hours or minutes. The beat rate in most clock timers is in beats per hour or BPH. In order for a clock to keep time, the pendulum length must be matched to the gear ratios of the movement to achieve a correct beat rate. The beat rate for a specific clock to keep time is a constant no matter where you are in the world, provided the gearing hasn't been modified from its original design. Using a timer to make adjustments to the pendulum length to get the correct beat rate is a fast, easy method and can be very accurate

Variations in acceleration due to gravity (g) also play a role in determining the length of a pendulum, and it is important to understand how that relates to beat rate. The variation in g is greatest when traveling in a North-South direction rather than in an East-West. 65% of the variation when traveling North-South is due to the rotational affect of the earth. This will have the effect of counteracting g. 35% is due to the Equatorial Bulge which changes your distance from the center of the earth[1] (see table 1). Other affects on g are changes in altitude (see table 2), and local variations due to the Earth's non-uniform surface density. As you move about the earth, g increases or decreases. This means that a pendulum will have to be lengthened or shortened in order to obtain a correct beat rate. It is for this reason that we must consider what the French used as the value for the acceleration due to gravity (g). Was it the value of g in London, Paris, the equator, the poles, or somewhere else? We must assume that the numbers stamped on the back plate represent a specific measurable length, and that the length was calculated using known equations which rely on the value of g.

            621.321 miles = height of a satellite                        236,121.056 miles = height of the moon

So, what does the value of g have to do with determining beat rate?
The value for acceleration due to gravity (g) is derived from Newton's second law of motion:

F = ma
            where:  F = force, m = mass, and a = acceleration
            Let acceleration (a) equal acceleration due to gravity (g).  a = g.

Then F = ma becomes:

F = mg

Now consider Newton's law of gravitation:

                m₁m₂
 F =  G   ^______
                  r²

            Where:  F = force between two particles having mass, m₁ = mass of particle 1,
                        m₂ = mass of particle 2, r = distance between the two particles, and
                        G = universal gravitational constant = 6.6726 x 10^-11 m³/kg s²
                                    (the above value for this constant was adopted in 1982 and first accurately measured by Lord Cavendish in 1798)

Letting m₁ be the mass of our pendulum (m), m₂ be the mass of the earth (M_e), and r the radius of the earth (R_e), we get:

                mM_e                GM_e
 F =  G   ^______  =  m   ^______
                  R_e²                   R_e²

Therefore, from Newton's second law of motion F = mg, we see from the above equation:

         GM_e
g =   ^______              which is not a constant and varies with location.  (see table 1 & 2)
          R_e²

Where:

G = 6.6726 x 10^-11 m³/kg s²
M_e = 5.97 ^. 10²⁴ kg  or 6.6 ^. 10²¹ tons
R_e = 6.37 ^. 10⁶ m  or 2.09 ^. 10⁷  ft
(1982 values)

In physics, the period of oscillation of a simple pendulum (T) is equal to two times pi (p), times the square root of the pendulum length (PL), divided by the acceleration due to gravity (g).

T = 2p SQRT (PL/g)

Here, "period of oscillation" of a simple pendulum is taken as the time for the total angular travel of the pendulum.  (see fig. 2)

This would mean that one beat is half of the period of oscillation since an escape wheel tooth receives a blow twice in each period of oscillation.  To make matters more confusing, this half-period is also widely used as T.  For clarity, I will use T_1/2.  Then we have:

T_1/2 = p SQRT (PL/g)

If we let T_1/2 = 1 sec. per beat and g = local acceleration due to gravity, the length of a seconds pendulum can be determined at our location.  (see table 3)

1 sec/beat = p SQRT (PL/g)

Inverting we get beats per second (BPS):

                     SQRT (g)                           60 ^. SQRT (g)
1 BPS =  ^__________       or     1 BPM =   ^__________
                      pSQRT (PL)                                 pSQRT (PL)

Squaring both sides gives:

                     3600 ^. g
1 BPM² =   ^__________
                          p^{2 .} PL

                                                                     3600 ^. g
Now, let  the pendulum constant P_c = ^__________
                                                                          p²

Then:

                                    P_c                                                    P_c
1 BPM² =   ^__________      or     PL =   ^__________
                            PL                                         BPM²

Which are dependent on location because P_c is dependent on g.

Converting to BPH:

                        3600 ^. P_c
1 BPH² =   ^__________
                           PL

And finally:

1 BPH = SQRT (3600 ^. P_c/PL)                     

We can derive a beat rate from the equation based on pendulum length and local pendulum constant,  provided the numbers used for PL and P_c are accurate. 

Remember, the purpose of this table is to aid in rating a clock.  If the clock doesn't rate at the suggested BPH then there may be other factors involved such as altered, damaged, or switched parts.  What counts is that the minute hand goes around the dial once every 60 minutes for the duration of its run. 

References:
[1] Fundamentals of Physics Second ed.; Halliday and Resnick; John Wiley & Sons, New York; 1981; pp. 218 - 254.
[2] Handbook of Chemistry and Physics, 5th ed.; The Chemical Rubber Company; Cleveland, Ohio; 1916; pp. 244 - 249.
[3] The Modern Clock; Goodrich; North American Watch Tool & Supply Co., IL; 1905; p. 12.

Method for Setting Clock Accuracy
Important note: You can try using the pendulum equation.... but you shouldn't, it will prove to be difficult in practice. You will need the accurate length L which is measured from the top attachment point to the end of the arm (not the end of the weight) and that usually involves taking the clock movement out of the case. If you take the clock movement out, you will usually have to adjust everything when you put it back in - chimes, clappers, gongs, alignment, weights, etc. It's just not worth it and you could break something, use the steps below!!

1. Most pendulums have their weights near the bottom of the length of the arm. To start the clock, gently push the pendulum arm laterally (to a side) and release. Let the pendulum swing about 10 times to reach a steady state oscillation.
2. Watch or listen to the pendulum as it swings while watching a modern watch. You should be able to tell if it swings "mostly" on time, once a second. Only adjust the weight at this point if it is obviously much faster or slower than your modern clock.
3. Pick a consistent measuring point: the instant when the chimes begin on the hour. When the chime begins, start a stopwatch
4. Wait just under an hour, and be ready for when the chimes begin. When they begin, stop the stopwatch.
5. Determine how far off the pendulum clock is from the stopwatch. Record the time as T1. If the stopwatch shows less than an hour passed, move the weight lower. If the stopwatch shows more than an hour passed, move the weight higher. Most arms and weight systems have an adjustment, often a screw. If you turn the screw or other locking device, make sure the weight moves too!
6. Pick an amount to adjust, like 1/8 inch, or 4 turns of the screw, or something similar.
7. Repeat steps 3 and 4 and get another time measurement. Record the time as T2. Now, you have a measurement, an adjustment, and another measurement. The difference between T2 and T1 is how much your adjustment affected the time!!
8. Determine how much time you are off, divide that by the adjustment time you just calculated in step 7, and then go ahead - move the pendulum that many increments to get to the time period you need! Then you'll need to do fine tuning again with the final adjustment. As you go about your business, time the one hour spacing and make fine adjustments to the pendulum to get the clock accurate.
9. Fine Tuning: this process takes days. I have tried with my pendulum clock to get it to be as accurate as a digital clock and I just can't and that doesn't surprise me. The best I have done is about 30seconds per day out of perfect accuracy. I set it accurately one day, then wait a day and measure it again. If it's a little slow or fast I move the weight screw one turn or less, reset the clock, and wait another day. Putting the clock more level helps.

You'll have to do this by trial and error, mostly; all clocks are different, and just maybe elevation will begin to factor in! But in general, after just a few hours, you will know pretty well how far the pendulum will have to be adjusted to set the time correctly. If you determine that the weight has to be moved beyond the length of the arm, then you'll have to contact a clock repair service, the spring may be worn out, but remember: adding more weight won't help!!

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