This page is a summarization of a Fall 2011 semester project intended to provide experience in the design and execution of an engineering experiment. It concerns the design, fabrication, and subsequent use of a Kater's Pendulum in finding the value for acceleration due to gravity to a high degree of accuracy. It is being publicly posted as a consequence of my realization that all information concerning the actual design of the instrument is unavailable or withheld. The following should prove useful to anyone seeking to design their own Kater's Pendulum or to at least understand it on a more fundamental level.
Determination of the Local Gravitational Constant by Method of Kater's Pendulum.
The goal of this experiment was to produce a reliable and precise value for the local acceleration due to gravity by means of a special mechanical pendulum, a photo-gate system for period timing, and an accurate form of distance measurement. This particular type of reversible pendulum, invented in the early 1800’s by Physicist Henry Kater, was employed as precision gravimeter. Having no record of the original pendulum’s design, the instrument has been re-engineered and the experiment conducted under the aid of modern measurement devices. With its use, the local gravitational constant at sea level in San Francisco, CA, has been determined to be 9.7955 [m/s2] which falls within 0.049% of the accepted value of 9.8003 [m/s2].
It should be noted that pendulums of very simple design are often used to determine the gravitational constant (colloquially fixed at 9.81 m/sec2, or 32.2 ft/sec2), although with significant intrinsic error. The primary benefit of Kater’s Pendulum is that it removes the causes of these errors, derived from hit-or-miss measurements of the intangible properties of the compound pendulum -- the location of the center-of-mass (C.O.M.) and the moment of inertia. Uncertainties in those quantities are large sources of error.
Consider a heavy ball hanging on a string. This type of pendulum mimics what is called a 'simple pendulum' in physics. The ball is treated as a 'point mass' of infinitesimally small size, and the string is treated as being weightless. Of course to treat any real system as such is certain to produce unreliable results. A more realistic pendulum would consist of a solid with a definite cross section and length, such as a bar of steel to be hung from one end. This is called a 'compound pendulum.' The problem with this is that the equation relating the gravitational constant to the observed motion of the pendulum would require absolute knowledge of its mechanical properties. This is impossible as the bar of steel surely has minute imperfections that are difficult to account for in calculations, much less to measure.
Enter the Kater's Pendulum. For any compound pendulum, British physicist and army captain Henry Kater proved in 1817 that there could be a second pivot-point located opposite the center of mass from the first pivot, from which the pendulum can be suspended to yield the same period. Should this unique condition be met, the Kater’s Pendulum can be modeled as a simple pendulum (point mass) of the same total mass and with a length equal to the distance between the pivots. This result places no dependence whatsoever on the location of the C.O.M. or the moment of inertia. Thus, all error due to geometric inconsistency is removed.
As the Kater's Pendulum is being designed, ironically, it is most expeditious to assume all components are of perfect dimension and density. Then, to accommodate the inevitable inconsistencies with this assumption, a moveable weight is designed to lie between the pivots in order to impart adjustability to the mass distribution. This ensures that the two periods can be brought to near equality. Once this condition is met, only two measurements are needed: the value of the ‘equalized period’ and the distance between the two pivot-points. The derivation of this result first requires finding the period of a compound pendulum:
For now, we assume that this compound pendulum has been properly designed such that there does exists a second pivot-point near the other end of the pendulum’s shaft, on the other side of the C.O.M. which would yield the same period. It will now be shown that the equation for the period of this special type of reversible compound pendulum can be greatly simplified to resemble that of the ideal simple pendulum. This is the derivation of Kater's result:
Additional considerations are available to improve the accuracy of the result. For instance, error introduced by treating the actual angle of oscillation as a ‘small-angle’ is significant and can be largely removed by applying a correction factor to the measured period:
Finally, it is should be apparent that during experimentation, it will be not be possible to get the period of the pendulum as hung from pivot A to equal that from pivot B. Friedrich Bessel showed in 1826 that any slight inequality in TA and TB could be accounted for by approximate knowledge of the relative position of the two pivots with respect to the C.O.M. This result, an alternative to the boxed result for g above, is given below. The experimenter should note that it requires finding the location of the moveable mass after the two periods are brought as close to equal as possible. This location is determined by balancing the pendulum, as the last step, upon some edge, and recording the distance to one of the pivots. This measurement can be done without too much fuss, as the consequence is merely a minute discrepancy in the 'fraction of differences' seen in the denominator below, whose value is quite small with respect to its partner on the left.
A surprisingly large challenge was found in designing the instrument. In short, the fixed components of the pendulum must be chosen such that, during experimentation, a smaller moveable mass of reasonable weight would be adjusted along the shaft to some reasonable final location. Although there are infinite solutions to this problem, most solutions do not satisfy these design constraints.
The approach to solving this problem was iterative in nature. First, the shaft of the pendulum was chosen to be a length of aluminum bar which would accommodate two pivots spaced by 1 meter (roughly translating to a 2 second period). Next, the large fixed weight was selected as (2) steel disks, permanently bolted to the shaft and located very close to one of the pivots, opposite the C.O.M.
(Of important note, the fixed weights are not necessary! They do however, pull the C.O.M. away from the center of the pendulum, which creates a large difference between rA
. The effect is that Bessel's function can be instituted using a comparatively crude measurement on the location of the C.O.M. If you do not intend to try Bessel's formula, than you may omit these weights!)
The design's only unknowns are both the smaller moveable weight’s mass and location. Arbitrarily, the location was chosen to be ¼ of the distance between the pivots, leaving only the mass to find. The ensuing calculations are not difficult, but are tedious, and so they were carried out either by hand, in Microsoft Excel, or using Wolfram Mathematica. Generally speaking, expressions for TA
were written in the terms of the unknown mass (m) and were set equal before solving for the value of (m) which satisfies. Design parameters (component sizes and masses, i.e. shaft material, fixed weight diameter or location, etc.) were adjusted, and the calculation repeated until the moveable mass was reduced to a reasonable size. These calculations are shown below.
Originally intended by Henry Kater to be made of steel knife-edges resting against agate plates, the bearings of this pendulum were refined to be more conveniently obtainable. 'High Speed Steel' lathe bits which were fixtured to rest over the edge of a sturdy table served at the 'male' half of the pivot. Two hard bronze bushings, pressed into the pendulum shaft, served as the female component. In this way, the pendulum is easily reversed, and when set swinging, is subject to minimum friction.
Again, the design approach begins by choosing a reasonable material, size, and location for the shaft and fixed weight. The following calculations determine the mass of the moveable weight, given that its location is assigned to be 1/4 of the distance between pivots, towards pivot A:
Thus, the mass of the moveable weight should have mass of m=0.617 kg (1.36 lbf). Ultimately, this will give the pendulum a mass distribution such that the periods can be made to be equal if suspended from either pivot. Kater’s formula for the gravitational constant will then become valid.
It should be noted that the other solution, m=4.66 kg is discarded. In fact, many such solutions will arise, including negative or complex conjugate masses, and can be avoided by altering the location or size of the pendulum’s shaft or large fixed mass by means of trial and error.
The other devices used in the experiment: a photo-gate timer and a lathe (used essentially as a
giant ‘caliper’) were obtained in hopes of increasing the number of significant figures in our
measurements. The photo-gate timer, for example, had a claimed accuracy of 0.05% with microsecond
resolution, and the digital reading on the ‘caliper’ had an accuracy of 0.001” and resolution of 0.0005”.
(2) Cast Bronze Sleeve Bearings: 0.750” OD, 0.500” ID
(1) 6061 Aluminum Bar: 1.23m x 0.038m x 0.01m (48.38” x 1.5” x 0.25”), for shaft.
(2) Large Steel Disks: 0.15m (6”) OD, for fixed weights.
(2) Small Steel Blocks: 0.12m x 0.07m x 0.01m (4.75” x 3” x 0.5”), for movable weights.
Now that the mass for the small moveable weight is known, steel is chosen as the material, and the volume is determined. The other components were finalized prior to the calculations above -- an aluminum bar for a shaft, two steel disks as fixed weights, and two cast bronze sleeve bearings to serve as bearing surfaces. The mating bearing fixture, to hold the High Speed Steel (HSS) bearing edge was designed and machined on the fly by sandwiching the HSS bit between two pieces of aluminum flat bar. All the materials required are pictured below.
We had two HSS bit and bronze bearing sizes to work with. We opted for the smaller set since the overall weight of the pendulum was relatively low at seven pounds. An added benefit was that the smaller bronze bearings replaced nearly the same mass of aluminum that was removed during drilling A combination square, scribe, and punch was used to first layout the (4) holes to be drilled along the shaft. Two of those holes were merely clearance holes for the 1/4-20 bolts which fixed the steel disks to one end. The other two holes were first drilled at 47/64" diameter and reamed to 0.7500" diameter. The outer diameter of the bearings, measured at a slightly larger 0.751" diameter was intentionally installed as a press-fit. The ensured a permanent arrangement, with the bearing axis quite perpendicular to the pendulum's shaft.
The smaller moveable weight was machined such that a specific volume is generated. Of course, it needs to arrive at the required mass (or at least close to it), so the material density is an important consideration. Our design sandwiches the pendulum shaft between two steel sections with enough room inside such that it slides freely. Its location can be fixed, however, using an added pair of set screws (not visible in photo). Our final mass of 607g is sufficiently close to the required 616g.
The 'male' part of the bearing required a rigid fixture which would present the HSS bit with an edge facing upwards. For this, we sandwiched the bit between two strips of 3/8" thick aluminum, bolted tightly together. The bit measures 1/4" x 1/4" x 3". The entire assembly is designed to easily clamped to a table or surface.
With construction complete, we tested the apparatus. An initial test run using a cellphone timer, and with the moveable weight in the previously predicted location along the shaft, yields a period of 2.0 seconds. We flipped the pendulum over and hung it from the opposite pivot. According to the cellphone, the period was also 2.0 seconds. We knew were were on the right track.
It is extremely important to know the exact distance between the pivot points. Our layout technique, using hand tools and punch-marks could not be trusted to more that +/- 1/16". Historically, traveling microscopes were used to measure this distance. Fortunately, the pendulum fits within the travel of the larger lathe. This lathe is also outfitted with a digital readout (DRO) on the axes.
The DRO has a resolution of 0.0005" and is an excellent tool for our unusual purpose. The pendulum is shown above mounted between the lathe's headstock and tailstock.
The cutting tool (chosen to be a parting tool for its flat edges and thin profile) is used to 'touch-off' against gauge pins which have been set inside the bronze bushings. The 'touch-off' process is very delicate and accurate.
Above, the cutting tool is slowly advanced towards a 0.6250 gauge pin (which has been snugly inserted into the bearing) until it can hold the weight of a 0.005 steel shim between the surfaces. When this point has been reached, the DRO's Z-axis origin is established or 'zeroed,' and the machine can be moved toward the opposite bearing. (Note that tool is actually 0.6250"+.005"=.0630" from the actual pivot. This will be taken into consideration later)
Now, at the other end of the pendulum, the process is repeated. First the tool is advanced (from the right side again), then a shim is inserted as the surfaces become close.
The lathe carriage handwheel is advances agonizingly slowly until the two surfaces barely support the weight of the shim. At this point, the location of the tool relative to its origin is recorded.
The number shown below is the distance traversed in the Z-axis between the two 'touch-offs.' The thickness of the shim can be neglected since the right side of each pivot was used. Additionally, the diameter of the gauge pin must be added. Thus, the distance between the two pivots is measured as 39.2770" Note the deviation from the nominal 1 meter.
With the design calculations completed, the machining complete, and the distance between the two pivot points accurately recorded, the experiment could begin. We clamped the male pivot fixture to the rigid fork of a heavy forklift, and began to assemble our experimental setup. We laid a precision machinist's level across the male pivot edge to check that is was completely horizontal.
The setup was simple. A photogate timer is arranged with its light gate at the lowest point of the swing of the pendulum. Because the pendulum can hang from either of its pivots, its lowest hanging point differs in distance from the ground by a few inches. This distance was taken up by supporting the photogate by a 'lab-jack' when necessary. The photogate was an older, but very nice model: Pasco Scientific Model 8025, which sports a one microsecond resolution, and a 0.05% accuracy.
Because the initial derivation of the governing equations requires use of the small-angle approximation (to preserve linearity), we try to adhere to this limitation during experimentation. As such, we have chosen to use a 2 inch lateral displacement of the end of the pendulum to enact swinging. This results in approximately a 2.8 degree angular displacement. Regardless, we will correct for non-linearity later.
The moveable weight was moved to the theoretical final location, and the pendulum was set swinging. For each location of the moveable weight, we recorded (3) periods using the photogate timer, before reversing the pendulum and recording (3) more from the other end. In this manner, we continually adjusted the location of the moveable mass until the periods became equal.
Each time we fixed the moveable weight at a location, we recorded its distance from the end of this pendulum. Although this measurement plays no role whatsoever in future calculations, we felt that we would gain a better understanding of the system by later recording how the pendulum's periods are related to the mass distribution. As we already knew, it was a complex relationship. For instance, by moving the weight towards one end, it may be the case that BOTH periods increase (although one might increase less than the other). It was very intuitive. Eventually, we arrived at a location where the periods were nearly equal.
To summarize the process:
1. Clamp the male pivot to a very sturdy table.
2. Move the moveable weight to some point on the shaft, and tighten it there.
3. Measure the location of the movable weight (just for fun).
4. Hang the pendulum on pivot A.
5. Displace the lower end of the pendulum by 2 inches and let swing.
6. Record the period (3) times using the photogate.
7. Flip the pendulum over and repeat from pivot B.
8. Move the movable mass to a new position.
9. Repeat from Step 3 until the periods are equal from both pivots.
After finding the location where both pivots where equal, we explored other locations on the pendulum's shaft to see what would happen. To our surprise, another location near the opposite end of the shaft yielded the same period. During the design phase, had we instead fixed the moveable weight's mass and solved for its location, we would have found this as a second solution beforehand. We did not spend much time there however, as we had already 'equalized the periods' at the earlier location.
The pendulum was securely fastened to a rigid and heavy surface, and the pendulum's period was recorded under small displacement oscillation. This process was repeated continuously, generating numerous sets of (3) period measurments for any given location of the moveable mass. After the location had been reached for which the periods were equal, other locations were quickly examined along the entire length of the pendulum's shaft, and the experiment was ended. The entire process was completed at approximately 10 feet above sea level in San Francisco, CA, USA on 11-19-11. The temperature was around 60-65 degrees Fahrenheit, with unknown humidity (clear cool day). The recorded data, that is, the periods about Pivot A and Pivot B, are respectively:
To clarify, the "distance" column is a measurement on the location of the moveable mass, arbitrarily taken from the end of the pendulum nearest Pivot A. The purpose is merely to assist in the generation of the following plot, illustrating the dependence of the two periods on the weights location. As can be seen, it is a perhaps unexpected relationship:
At the first point of period equalization, occurring when the moveable weight is approximately 9.16" from the end of the pendulum, a measurement on the location of the center of mass at that point must be taken. You need to know where the center of mass is here because Bessel's special function requires its knowledge (although the general result of Kater's pendulum period formula does not).
If you look closely at Bessel's function (see top of page), it should be clear that this measurement need not be done to an agonizing degree of accuracy. For this, we simply lift the pendulum and rest it horizontally on the male pivot edge. We move it incrementally back and forth until it balances, and we took a measurement with a large pair of calipers. The measurement was taken as the distance from one of the pivots to the bearing edge of the male pivot. This number was ####,"