The Pendulum by Shelby Cain  and  M Nealon
 An idealized pendulum oscillates over small amplitudes only and consists of a bob attached to a massless rod.  The rod constrains the bob to move along a circular arc.  Confining the motion of the pendulum to small amplitudes allows for a simple expression for the period, τ; namely,      Length is measured from the rod’s pivot point to the center-of-mass of the bob regardless of the size or shape of the bob, or method of suspension.  The acceleration due to gravity at the earth’s surface, g, varies with location because the earth is an oblate spheroid of non-uniform density.  Here, we take g to be 9.80665 m/s2, but neither the number of significant figures nor the exact value are important as long as the value does not change wherever the pendulum happens to be.   Thus, for a two-second period, length = g/π2 = .993621 meters.  A two-second period is quite convenient for clock making since the clock would advance one second for each half-swing of the pendulum.  This arrangement may decrease the complexity of the timing chain of gears driving a mechanical secondary movement – if one is used.   Notice that of all parameters that might describe a pendulum, the period depends on length alone.  If the pendulum regulates a clock then the time rate can be adjusted by a simple adjustment of the pendulum’s length.   The accuracy of a pendulum clock is highly dependent on the exact adjustment of the length simply because there are a lot of seconds in a day.  Suppose the length was misadjusted – too short, say, by 0.1 millimeter (about four thousandths of an inch); that is, suppose the length was 0.993521 meters.  If this were the case, the pendulum would have a period of 1.99990 seconds.   As there are 86,400 seconds in one day, this pendulum would oscillate       A properly adjusted two-second pendulum oscillates 43,200 times in one day, so this improperly adjusted pendulum has two extra oscillations.  Two extra oscillations per day means the clock ticks four extra times per day, or an error of 4 seconds per day “fast”.  Four seconds per day is about the same as 2 minutes per month, or about 24 minutes per year.  This might not be acceptable for a clock.   However, measuring the pendulum length this precisely would be difficult even with precision measuring instruments such as calipers or micrometers.
 In a practical sense, a clock’s time rate is not set by the direct measure of the pendulum length.  The length is adjusted as precisely as possible and the rate is determined by watching the clock over an extended time.  The rate is then compared with some outside reference such as radio station WWV operated by the National Institute of Standards and Technology (the old National Bureau of Standards).  The length is then readjusted as necessary.   A real pendulum deviates from the ideal in a number of ways, some of which may be compensated for, some may not.  For example, with a spring steel suspension the precise location of the pivot point may not be fixed in place; that is, the effective point about which the pendulum rotates may shift downward slightly as the spring flexes, thus in essence altering the length of the pendulum.  This inability to account for all aspects of the real pendulum means that these calculations should be taken only as a guideline in the design of a pendulum.   For a practical clock, considerations such as the rod length and its material, the mounting brackets, the size of the cabinet, the gear ratios of a secondary movement, the driving mechanisms for both the pendulum and the secondary, the spring suspension, the bob design, the energy lost in deflecting the air molecules, any changes in humidity, atmospheric pressure and temperature, etc., ought to be taken into account before construction begins.   The pendulum must be driven in such a manner that any energy loss due to friction at the pivot, and due to air resistance would be overcome exactly by the driving force; if so, a constant amplitude results.  In other words, the pendulum must be driven at a constant rate over a long time compared to the period.  Furthermore, since the period (of the ideal) is determined by the distance from the pivot point to the center-of-mass, the length must be constant.   For circular-arc motion, the period is not isochronous but presumed so for the derivation.  The period of a pendulum moving in a circular arc depends, however slightly, on the amplitude – in other words, a larger amplitude has a longer period; this is a second-order effect and is called “circular error”.  A proper isochronous pendulum would follow a cycloidal-arc.  A pendulum with circular-arc over a small amplitude is a good approximation of isochronous motion, and, in addition, is easier to make than a cycloid.  It is true, of course, that a pendulum with circular error, however large, may be used equally well as a timekeeper – as long as the amplitude is held constant.     The EOS Physics Laboratory Pendulum   In considering a real pendulum, the length will vary if the rod expands or contracts with changes in temperature.  A few standard techniques have been developed over the years to counter the problem associated with thermal expansion.  This problem may be lessened by using a rod with a low expansion coefficient; however, these rods may have substantial mass and perhaps cannot be approximated as massless.  Likewise, since the bob itself will expand or contract with temperature changes, a slight change in the effective length of the pendulum may occur.  One old technique is the use of glass vials of mercury as the bob.  These vials are suspended at their lower end so that the liquid mercury expands upward in the vial; this compensates for the downward expansion of the rod.  Mercury is selected because it has a relatively high thermal expansion coefficient.  Another technique, used here, is to suspend the bob at its center-of-mass.  In this manner, the bob’s expansion or contraction about its center-of-mass has no effect on the period.  This is illustrated in the figure below.
 Support of bob at its center-of-mass, in cross-section view. The rating nut is used to adjust the position of the bob and thus the length  of the pendulum system.  The bob is supported at its center of mass by a sleeve over the  rod.  First, the sleeve length is calculated to counter the thermal expansion of the rod; the total bob height is then designed to be just shorter than twice the length of the sleeve.  The sleeve sits on the rating nut.  Note that in order that the threads engage the rating nut, and to allow for a little leeway in making the adjustment, the rod must extend a small distance beyond the sleeve. ____________________________   The bob is a cylindrical brass can, 6.06” high x 3.00” diameter, filled with lead shot and with end caps fully soldered on.  Its mass is 5.9665 kg.  The central tube, through which the rod passes, is constructed with three parts, also in brass: in the bottom-half of the bob, which accommodates the sleeve, the central tube has a 13/32” inside diameter; the upper-half tube has a ¼” inside diameter.  The two tubes are joined together by a small shelf, or platform – the bottom of which rests on the sleeve.  The sleeve used in these experiments, 3.17” long x 9/32” outside diameter, is brass, although the bob is designed for a copper or stainless steel sleeve.   The rod is Invar steel, ¼” diameter.
 Pendulum mount   The spring steel used as a pivot is ½” x 1” x 0.005” and is supported at the top by a brass machine screw which passes through a hole in the steel spring.  Knurled nuts act as clamps.  A flat brass piece in the shape of a “T” is silver-soldered into a slot in the top of the Invar rod.  This “T” engages into a copper hook (attached by #4-40 machine screw) at the bottom end of the spring in such a manner to allow for a quick disconnect of the rod without screw fasteners or other clamps.   Center-of-mass   Pendulum rods are not really massless so we may wish to include the mass of the rod as part of a “pendulum system”.  The rod’s mass has the effect of shifting the system’s center-of-mass upward in comparison to an ideal massless rod of the same length.  The position of the bob must therefore be shifted somewhat, compared to the ideal, in order that this real pendulum have the desired period.   For a two-second pendulum, the center-of-mass of the system must be 0.993621 meters.  Let us call this length Xtruecm.  The purpose here is to calculate the position of the center-of-mass of the system, as illustrated in the figure below.  The origin of our one-dimensional coordinate system is the pivot point. For two objects, in this case the bob and the Invar rod, the position of the center-of-mass is   where xr is the distance from the pivot point to the center-of-mass of the Invar rod, xb is the distance from the pivot point to the center-of-mass of the bob,  mr is the mass of the Invar rod, mb is the mass of the bob.    The distance, Y, from the pivot point to the top of the bob is measured with an ordinary meter stick.  Once this distance is determined, the length is not adjusted further during these experiments.  We find
 Sonic detector data from Logger-Pro software by Vernier.   A sonic motion detector and curve fitting software have been used to determine the period.  This is a typical run with no coins on the bob.  Although only 30 seconds of data are shown here, the length of all data runs was 60 seconds; the rate at which data was taken was 20 samples/second; the best-fit sine function is in black and the data is in blue.  The amplitude is not such a great fit, but the period is     Period data taken about one week apart.  Shown in red is the mean τ value, 1.9992702 seconds.  Typical RMS error bars are given, calculated by the Logger Pro best-fit sine function to sonic detector data.   Include the Small Parts   At this point it might be interesting to include in the true center-of-mass calculation, all the constituent parts of the pendulum just to see whether we are justified in excluding them.  The most massive among these small parts are the magnet and rating nut.  The mass of these parts have the effect of increasing the effective length of the pendulum.  With the small parts, then, the center-of-mass measured from the pivot point is     = xr = 0.5602 m xb =1.0015 m xspring = 0.00127 m xnut =1.0852 m xmagnet = 1.1336 m xsleeve =1.0416 m xspringclamp = 0.00254m mb = 5.966 kg mr = 0.223 kg mspring = 0.00042 kg (average of several springs) mnut = 0.02262 kg mmagnet =  0.02425 kg msleeve = 0.00715  kg mspringclamp = 0.00357 kg.   Consequently,   Inclusion of these small parts changes the value of the effective length of the pendulum by an amount about one-half millimeter.
 One-dimensional thermal expansion of a rod of length L0 may be described by the empirical formula   where, ΔL is the change in length, α is the thermal expansion coefficient, and, ΔT is the change in temperature.  The rod and sleeve are to expand an equal amount for a given temperature change, so we can write   Let us pick a material,say, copper, for the sleeve and calculate the length required.  To calculate the sleeve length we need the length of the rod from the pivot point to the rating nut.  The distance to the center of mass of the pendulum system is very close to one meter exactly; to this we must add half the height of the bob.  This total length is 1.085 meters.  Therefore,   Thus, the bob is 6.1” in height.